8.2: Installation Art
Introduction
Installation art is a relatively new genre for the millennium defined as large-scale constructions, usually mixed media, and installed for a specified period. The artwork usually fills the space, and viewers must walk through the exhibit, often becoming part of the installation. However, some installations might be fragile or installed along a wall. A sculpture is generally one piece of art displayed in separate or individual spaces. An installation is a more unified experience, engaging the viewer in most of the environment. Technology has also contributed to the ability of artists to create large installations involving LEDs, computerized movement, unusual formations, and even environmentally supported projects. One attribute of installation art is its immersive properties, how the art appears depending on the viewer's position. The works are also large in scale and site-specific, made to fit into the gallery and museum. or outdoor space. Generally, they are placed in unique surroundings to compliment the artwork.
Ai Weiwei
Ai Weiwei (1957-) was born in China, his father a poet. The family was sentenced to a labor camp when Ai was one, then sixteen years in exile in far northern China. After Mao Zedong's death, the Cultural Revolution ended, and the family moved back to Beijing, where Ai studied animation. He moved to the United States for twelve years to learn English and art. He returned to China when his father's health failed in 1993 and set up his studio. Ai began to openly criticize the Chinese government's human rights issues, corruption, and lack of democracy. Most of his work follows his political beliefs and feelings about the social problems in China, leading to harassment by the government and occasional house arrest. In 2011, Ai was jailed for three months; however, the government kept his passport when he was released, and he could not travel. When his passport was returned in 2015, he moved to Europe. Ai Weiwei uses his art as dissident art, protesting human abuses in China by the government. His work is international and brings his messages to the world. He uses simple or everyday objects or materials to create his installation as he believed they were meaningful to people.
Sunflower Seeds (8.2.1) is composed of 100 million small seeds stacked evenly on the floor. Each seed appears the same; however, every seed is unique. The life-sized seeds are made by hand from porcelain, sculpted, and painted. The seeds were produced in small workshops in the city of Jingdezhen, known for their production of porcelain since 1004 CE, and originally produced the Imperial porcelain for the royal palaces. Revolutionary types of kilns and access to deposits of petuntse , the clay needed for porcelain were part of the cities success to become the major Chinese porcelain site, even today. Ai Weiwei used the concept of porcelain sunflower seeds to examine the idea of 'Made in China' and its position in the global economy. He created each seed piled in the installation to represent the relationship between an individual and the masses. He posed the questions: "What does it mean to be an individual in today's society? Are we insignificant or powerless unless we act together? What do our increasing desires, materialism, and number mean for society, the environment, and the future?" [1]
Ai Weiwei's installations reflect his commentary on society, politics, and economics in contemporary China, and he uses irony to create his installation. World Map (8.2.3) is an example of simple, basic objects to make a simple visual expression.
The map comprises thousands of layers of very fine, thin cotton cloth (8.2.4). Making, cutting, stacking, and installing the cloth was exceptionally time-consuming and laborious. The map demonstrates China's position in the world as a place of cheap labor, especially in the clothing industry. Ai stated, " China is blindly producing for the demands of the market… My work very much relates to this blind production of things. I'm part of it, which is a bit of a nonsense." [2] He also uses other materials to make maps of China in protest, including 1,800 milk tins to protest China's tainted milk, making infants sick.
In the installation, Moon Chest (8.2.5), Ai made 81 chests of wood, each with four circles cut in the upper and lower panels on both sides. The openings vary on each chest, and when the chests are carefully aligned, the moon's phases are visible. He wanted to create an installation of simplistic materials and construction while having a sense of functionality. The chests were made from Huali wood, a common material for Chinese furniture, and were constructed without any joining materials, the methods used for ancient furniture. The exhibition was one of the few of Ai's installations without a direct political message, emphasizing artistic beauty.
Ai Weiwei created different bicycle installations to acknowledge China's past reliance on bicycles, the mass-produced transportation now becoming obsolete to the car culture. Forever (8.2.6) is one of the installations. Bicycles can be installed in multiple configurations representing the universal use of the bicycles and the beauty found in the object. However, the chain and pedal have been removed from each unit, now obsolete and unusable. However, the bicycles are arranged in a circle giving the feeling of motion—the "forever" of the title belying the actuality.
Ai Wei Wei's accumulations and use of single objects are used again in installations with stools. Both installations (8.2.7, 8.2.8) demonstrate how he combined small and large sets of stools. The three-legged wooden stool was typical in China, now an antiquated object in today's Chinese society. Previously, the stools were carefully carved and passed down through the generations, an expression of centuries-old aesthetics. Today stools are plastic or aluminum in China, a modern and mass-produced object, disposable in today's society. A common symbol of the importance of and creativity of the individual as opposed to the massive state.
Yayoi Kusama
Yayoi Kusama (1929-) was born in Japan, where her parents owned a nursery for plants and seeds. Kusama started drawing pumpkins from the time she was a child. She said she drew from hallucinations she observed, a concept followed throughout her life. Kusama's childhood was traumatic; her mother did not support her art and was abusive. Her father had continual extramarital affairs, which her mother made Kusama follow him and spy, reporting back. By the time Kusama was ten, her hallucinations had included flashing lights and fields of dots she believed engulfed her. She studied traditional Japanese Nihonga style, finding it unsatisfactory, and studied Western avant-garde. After success in Japan, at twenty-seven, she went to the United States from 1957 to 1972, believing Japan was too disparaging of women and too feudalistic.
In the United States, Kusama developed her reflective mirror rooms, complex installations with mirrors, colored balls, and lights. She worked successfully with other artists in New York City, however financial success eluded her. During this period, Kusama became heavily involved with the anti-Viet Nam war protesters, often doing unusual things to draw attention to her demonstrations, including painting polka dots on nude protesters. When Kusama returned to Japan in 1973, mentally ill and suicidal, she checked herself into a hospital. Since that period, she has resided in the psychiatric hospital, creating her art in a nearby studio.
Dots have become Kusama's obsession on all her artwork and installations, dots covering everything in repetitive patterns. Kusama said, "A polka-dot has the form of the sun which is a symbol of the energy of the whole world and our living life, and the form of the moon which is calm. Round soft, colourful, senseless and unknowing. Polka-dots can't stay alone; like the communicative life of people, two or three and more polka-dots become movement. ...Polka-dots are a way to infinity." [3] Most of her work is based on the repetitive use of dots, beloved pumpkins, and mirrors in her installations. Dots Obsession (8.2.9) is like Kusama's hallucinations in childhood, believing her space was covered in patterns. In this installation, she used red for the base color and the highly contrasting white dots. The room was enclosed with mirrors, multiple lights, and large balloon-like structures, giving the viewers an immersive involvement.
Repetitive Vision (8.2.10) creates a startling strange, and disorienting environment. The black mirrored walls and ceiling reflect the mannequins covered in red dots. The mirrors present views of the front, back, and sides of the mannequins confusing the viewer, impossible to know the limits of the room or numbers of mannequins. The red dots on the floor, mannequins, and reflections add to the confusion of the room's boundaries. Kusama said, "A mirror is a device which obliterates everything, including myself and others in the light of another world or a gallant apparatus which creates nothingness." [4]
Late in her career, Kusama's infinity rooms expanded and included her trademark dots and pumpkins. The mirrored and lighted ceiling and walls infinitely reflect the dotted pumpkins in All the Eternal Love I Have for the Pumpkins (8.2.11). The illusion of the installation stretched the concept of ad infinitum. Kusama has placed rows of yellow pumpkins painted with strips of black dots, making the room's depth impossible to judge. The space reflects Kusama's obsessive-compulsive disorder for the repetition of polka dots enclosing the person in the room, the pattern wrapping around and surrounding the viewer.
In Japan, pumpkins are known as kabocha; an image Kusama spent hours drawing as a child. Pumpkins represented stability and comfort and were attractive in color and shape. By the late 1970s, she started to include pumpkins in some of her work, covering them with dots and incorporating them into the themes of her mirrored rooms. Pumpkins became major subjects by the new millennium, and Kusama made enormous sculptures. The Red Pumpkin (8.2.12) and the Orange Pumpkin (8.2.13) were made from fiberglass for installation on Naoshima Island in Japan. The oversized red pumpkin sits on the edge of a protected inlet. The sculpture is hollow inside, made from fiberglass, and covered with large black dots. The orange pumpkin with a different set of black vertical dots was installed on the end of a pier adjacent to the water. The pumpkin was dislodged when a typhoon swept the island in 2021, and waves took it into the sea. The sculpture cracked but was recovered and will be repaired.
Kusama also made immense pumpkins from stainless steel (8.2.14, 8.2.15). The metal is shiny and reflects light and images like her mirror rooms. The dots on the pumpkins were made with long-lasting urethane paint.
Do Ho Suh
Do Ho Suh (1962-) was born in Korea, the son of a famous artist who used traditional paintings in ink to create more abstract concepts. Originally, Suh wanted to be a marine biologist, failing to achieve the grades. He went to art school in Korea and earned a master's degree in art based on traditional Korean painting. Suh relocated to the United States in 1991, freeing himself from his father's influence and traditional art and studying at the Rhode Island School of Design. Here, he was interested in how space was used and how people interacted in different cultures. Suh explicitly studied houses' psychological and physical dimensions and the architectural structures.
Suh's major project was Home Within Home (8.2.16), a life-sized reconstruction of an actual home where he once lived. Suh also constructed a replica of his home in Korea, contrasting style and size inside the house. He used translucent fabrics to construct the houses, interior rooms, and different large appliances. He also displays hallways, doors, or specific rooms from various places he lived in other installation parts. Suh used a 2-D scanning machine to ensure accurate and exact specifications. Using the defined dimensions and outlines, Suh used a traditional Korean hand sewing method to make every part of the building or objects, adding precise details found in the actual item. The oversized home is designed for the viewer to approach the building and see through the walls before entering the door and walking through the spaces. The rooms in the building create a dreamlike environment; shelves, doors, fixtures, windows, stairs, or appliances are viewable while still looking through to other spaces.
The bathtub (8.2.17) and the toilet (8.2.18) are created in exact sizes and placed in the bathroom. Suh used steel wire to frame the bathtub and toilet to hold the fabric properly. For the larger structures, he used stainless steel armature.
Suh was raised in South Korea, the influence of North Korea and the ever-present threat in the lives of those on the Korean peninsula. The two countries were a display of conflicting social and political societies. The images of North Korean soldiers in precise and continued marching manifested in the concept of the control and collectivism of the state over the thoughts and activities of the individual.
The sculpture Cause and Effect (8.2.19) is made of 1,200 naked resin figures cast in different colors. Each small figure sits on the figure below it, portraying the relationship between the individual inside the collective or the concept of the individual supporting the larger society.
El Anatsui
El Anatsui (1944 -) was born in Ghana, where his uncle raised him. As a child, he drew on a chalkboard and later trained at the University of Science and Technology in Ghana. His early work was based on traditional wood carving and sculpture and teaching at the University of Nigeria. His first entry into the international art world was in Harlem, New York, in 1990 when he entered a show about contemporary African artists. Anatsui found his art medium in the streets of the local towns, discarded liquor bottle caps. After collecting thousands of metal seals from the bottles and bottle caps, he must smooth and flatten them to be more malleable and workable. For his sculptures, he uses copper wire to attach the pieces. His work combines the concepts of traditional African imagery and the ideas of abstraction. Anatsui's work is moveable and easy to configure in different ways. His work does not have a fixed form, and when he sends a sculpture to an institution, he does not send any instructions, allowing them to decide how to hang and drape his work. Anatsui said, "I am interested in textile because it is always in motion. Anytime you touch something, there is bound to be a change. The idea of a sheet that you can shape or reshape. It can be on the floor; it can be up on the ceiling; it can be up on the wall, all that fluidity is behind the concept." [5]
Anatsui noticed walls worldwide and how they close space and stop the eye from traveling further. He wanted to create a wall the viewer can see, though fluid and moveable. The term Gli meant will, however with a different inflection, the word might mean story or even disrupt. Gli (Wall) (8.2.21) is an oversized installation hung from the ceiling representing a wall. However, it is a transparent wall, most of the metal pieces formed into circles, creating visibility through each small opening. Anatsui connected thousands of bottle caps and collars with thin copper wire. The reflective metal shimmers, giving the appearance of curtains moving in the breeze instead of rigid metal.
AG+BA (8.2.22) is an example of Anatsui's selection and application of color. Each of the bottle tops was branded by different companies who bottled the liquor. Anatsui collected millions of the tops and sorted them by color in his compositions. In part of the sculpture, the red pieces were interspersed with gold and some black; the other portion is formed with gold and small amounts of gold and red. The bottle tops are relatively soft and easy to flatten. Part of this sculpture is hanging and draped, the pieces wired together; the sizeable golden section does not drape and is welded into a massive block. Anatsui started using bottle tops when he found three bags of tops. The liquor bottles could be recycled; however, no one recycled aluminum.
Drifting Continents (8.2.23) depicts a fragmented map of the world where the continents collapse into each other. The work displays the interconnectivity of the regions and the creation of restrictive, poorly defined borders. Wealth and gold flowed out of Africa for centuries causing economic disorder today. In the sculpture, the flattened gold tops flow down the walls, beautifully reflective and demonstrating the beauty of gold and why it is coveted. The discarded refuse of the bottle caps recreating the richness of piles of gold only the value is only in the art, not the materials.
Subodh Gupta
Subodh Gupta (1964-) was born in India; his father died when Gupta was young. He was sent to live with an uncle on a remote farm, where children went to school without shoes or roads. He attended the College of Arts & Crafts in Patna, a place without even a library. When he graduated in 1988, Gupta moved to Delhi, starting his career as an artist. He always loved steel and metal, from the dull brass plates the family used for eating to the replacement stainless steel plates. Gupta started using the everyday kitchen utensils used at home, a material he uses. Line of Control (8.2.24) is a sculpture based on the mushroom clouds developed from atomic bombs and represents the dismaying deployment of nuclear war. Nuclear weapons are always an issue between Pakistan and India and the potential of their use in war. The sculpture also represents geopolitical borders, limitations, disputed territories globally, and potential nuclear warfare potential. The work was constructed with twenty-six tons of stainless-steel pans, thalis, bowls, milk pails, tiffin boxes, and various utensils. Each element was welded together to assemble the sculpture.
Gupta has always been interested in the migration or displacement in India, even why they travel to other places, carefully noting the changes in Indian society. The Silk Route (8.2.25) comprises stacks of an abundant supply of pots, bowls, and plates carefully arranged on a giant conveyer belt. He presents the growth of the middle class and its constant need for more. The reference to the old silk road defines the influence and threat of globalization changing local communities and the spread of consumerism. The towers of tiffin pots (stackable containers) move along the installation much like India's rapid pace of change. The tiffin pots were a standard method of carrying food when Gupta was a child, the objects now having a different meaning. Gupta said, "My work is about where I came from, but at the same time, the expansion of the art world means that to a certain extent, everything is shrinking together, and you have to be aware of international discourses in your work." [6]
 The tree, Specimen No. 108 (8.2.26), was exceptionally complex to construct. Gupta first made the metal trunk with the branches before the utensils were individually welded to the tree. The tree is shiny with the beauty of an actual tree, elegant and graceful, combined with the ubiquity of common household elements hanging like fruit. The banyan tree is incorporated into Indian culture as sacred and a center of worship; however, the number 108 is meaningful in multiple religions. The imposing tree is a symbol for other meanings. The mystical formation of the tree is further enhanced by the sun bouncing off the steel and the utensils softly moving in the breeze.
Jean Shin
Jean Shin (1971-) was born in Korea, where her parents were professors. When she was six years old, they moved to the United States. Shin graduated with a BFA from Pratt Institute and an MS in History. Her artwork is made from cast-off materials she collects and forms into installations. She is not particular in her accumulations, including one sock or a broken ceramic, discarded lottery tickets, and even old pill bottles. Shin creates large-scale sculptures and needs large numbers of any one element. She believes the object in an installation may all look the same until closer observation reveals individuality and variety. Shin wants the viewer to continually shift between the group and the individuals within the group, some things more intimate and others appearing excessive.
Huddled Masses (8.2.27) connects the idea of the environmental waste of technology and the desire from society for more and more technological products. The sculpture is made of old cell phones, obsolete by the following year's model. Meters of old, no longer viable computer cables encircle the structure capturing the phones into piles of meaninglessness. The toxic waste now sits, planned obsolescence forcing the new technology to the detriment of the environment from the masses of unusable waste. Large forms jut out of the middle like the ancient rocks of Chinese art, the purity of natural, long-lasting stones of the past, now covered by today's pollution. Oddly, the sounds on the phones made to carry the noise and discourse of society are now silent in their obsolesce. The most significant part of the sculpture is 2.28 meters tall. Shin collected over 3,000 different styles of phones, some as long as twenty years ago, for the sculpture.
Chance City (8.2.28) was constructed from thousands of losing scratch-and-win lottery tickets, discarded as useless. People purchase them hoping to make money; the losing tickets reveal the unfulfilled dream someone had when they bought the ticket.  She does not use any glue when erecting the house of cards, only balancing one on top of another. Although the sculpture looks fragile, gravity and friction hold the cards in place. Shin believes, "Picking up your life and moving to the city and giving I all you can, your dreams may change-transform. But somehow, I think all of us retain that memory of something that they really wanted to do, and against all odds, are able to succeed." [8] The ticket may not bring instant riches, but our odds of success are achievable.
Anselm Kiefer
Anselm Kiefer (1945-) was born in Germany a few months before World War II ended. The city was heavily bombed, a difficult place to live. Kiefer attended the University of Freiburg, first studying languages before switching to art. In the first part of his career, he painted images based on themes from Nazi rule, the Holocaust, or spiritual poetry. Kiefer incorporated straw, clay, ashes, shellac, and lead in the mix of paint applied to the canvas. He used many of his painting concepts and techniques for the large installations. Seven Heavenly Palaces (8.2.30) was based on ideas from an ancient Hebrew writing. The seven towers were fourteen to eighteen meters high, and each weighed ninety tons. Kiefer used reinforced concrete for the towers. He inserted lead wedges between each level. The towers became a summation of Kiefer’s life-long themes of the ancient Hebrews, World War II atrocities, the ruination of Western culture, and the possibilities in the future. He also added large paintings to reinforce his philosophies. Visitors walk through the space to view the towers and paintings, moving between spaces and media. Kiefer added different meanings to each tower. On one tower, he placed an ark representing the ark believed to rest on Ararat. At the base of another tower are small sheets of glass labeled with paper to represent celestial bodies.
Breaking of the Vessels (8.2.31) is a massive installation weighing over seven tons. The bookcase is filled with forty-one lead book folios and interspersed with glass. The books reference Jewish culture and ancient mystical writings. A semicircular pane of glass is suspended above and is inscribed with Hebrew words translating to “Infinite Light .” According to the writings, the attributes of G-d’s light were divided among ten vessels that were not strong enough to contain them and broke, bringing the Divine into an imperfect world. [9] Kiefer frequently referred to the atrocities of the Nazis against the Jews. In front of the bookcase are piles of broken glass, referring to Kristallnacht (The Night of Broken Glass). In November 1938, Nazis and civilians shattered the glass windows in thousands of Jewish-owned businesses and synagogues. The Nazis destroyed and burned all the books they found in Jewish homes, synagogues, schools, or businesses. The books are made from lead, impenetrable, hidden. Kiefer frequently used lead in his paintings and installations, lead representing a crushing, brutal force. He described lead as “the only material heavy enough to carry the weight of human history.” [10]
Kara Walker
Kara Walker (1969-) was born in California, her father a painter and professor. Walker said she used to sit on her father's lap and watch him draw, always wanted to become an artist. She received a BFA from Atlanta College of Art and an MFA from Rhode Island School of Design. Walker lived in California until she was thirteen in an integrated neighborhood surrounded by civil rights activists. When she moved to Georgia, Walker was shocked to discover the amount of discrimination, including the Ku Klux Klan rallies, still occurring. The discriminatory practices of the south highly influenced Walker, and her art focused on the brutalities of slavery. Walker is known for her panoramic silhouettes depicting racism, and the realities of life slaves endured. Her work was primarily black figures backed by a white wall using paper, video, shadow puppets, or other projections. Walker has also branched out and created oversized sculptures. Her silhouettes in African't (8.2.32) are almost life-sized. Men, women, and children who were black and white are depicted in different scenes before the civil war. The scenes present a diverse tableau of the white population's terror, violence, and inhumanity to the black people held in bondage. Walker displayed the spoken and unspoken of violence and sexual abuse. Each image has a specific interpretation, the details of a disturbing view of race relations in the American south. Walker usually positioned her silhouettes strung out against the wall or running down a wall by stairs, spreading the work out to view each frame.
Event Horizon (8.2.33, 8.2.34) is placed by the lobby's central stairway. The silhouettes represent the Underground Railroad and the African American's struggles for freedom, the installation resembling an Asian scroll. The first image depicts the man releasing a shackled woman who loses her hold of the child, all of them falling into the dark abyss. Walker's work is very detailed; the small fingers, braided hair, or ragged hem of the dress are all visible in the black cutouts. In any of Walker's images, the people's faces are not visible; the people's emotions are built into their body positions and interactions. The strong context of the black silhouettes against the stark white background removes all ideas of gray areas to explain the sinister horror of slavery.
In 2014, the massive sugar factory in Brooklyn, New York, would be demolished. Walker designed a monumental sculpture to represent the exploitation of the people throughout the history of the sugar trade. The building was constructed in the late nineteenth century to store the raw sugar from the Caribbean before the sugar was refined and packaged; sugar was a luxury. Hundreds of thousands of slaves were taken to the Caribbean to work on the vast sugar plantations, building the sugar market on the backs of Africans shipped across the Atlantic. Sugar was also known as blood sugar, defining how humans bled through the exploited work in the fields or under the brutality of the overseers. Walker designed an immense sphinx-like woman, representing the building and the history of sugar. A Subtlety or the Marvelous Sugar Baby (8.2.35) is located between two rows of steel columns, the Sugar Baby appearing as a colossal goddess.
The sculpture was created from 330 huge polystyrene blocks and then covered with eighty tons of sugar—the sugar coating over the blocks allowing the seams of the polystyrene to show like quarried stone. When the Sugar Baby is approached from the front (8.2.36), her enlarged hands appear foreshortened, her pendulous breasts filling the space and her enigmatic face.
Walking around to the back of the statue (8.2.37), the viewer comes upon her buttocks jutting up from the shortened spine, thighs, and calves. "A powerful personification of the most beleaguered demographic in this country — the black woman — shows us where we all come from, innocent and unrefined." [11]
Around the Sugar Baby were thirteen small molasses-colored boys (8.2.38) standing about one and a half meters high. The statues were made from cast resin and covered in molasses or cast sugar, some light, others darker colors. The figures are carrying bananas or baskets, appearing to bring offerings, or they were only tired workers coming back from the cane fields. The whole installation remained in place for a little over two months. The building was open to the elements, and the weather eroded the sculptures before they were taken down.
Liu Wei
Liu Wei (1972-) is from China and graduated from the China Academy of Art. At first, he created realistic paintings before working on installations. He moved to Beijing and was associated with other artists who produced exhibitions protesting the government. Their video, photography, and performances contained pictures of corpses, cadavers, or other gross and visceral images. Their concept planned on repelling Western audiences revealing their resentment of Western powers. However, as China expanded its trade globally, the artists opened to Western investors. Since the new millennium, Liu has been creating models of cityscapes using unusual materials. The massive construction of cities in China inspired him to create installations based on the continual change, decay, and construction found every day in cities. Today, Liu creates his image digitally before assembling an installation, continuing to change them over time.
Liu's installation Love it! Bite it! is made from the unusual material of dog chews. He tried to create buildings from across Western history. The work was exceptionally detailed, with ornate columns, domes, towers, and cornices. However, most of them resembled ruins of the dystopia of fallen empires. Liu made the construction from dog chews after seeing his dog gnaw on its chew as it was crumbled with saliva and dirt. He observed the dog's lust for food like a human's hunger for power, the city the representative of that power. Love it! Bite it! (8.2.39) displays the falling coliseum, a church, and other collapsing buildings.
Love it! Bite it! (8.2.40) is part of the United States capitol building exhibit eroded by greed and time.
In Library II-II (8.2.41), Liu constructed four sections displaying another dystopian city with seemingly familiar buildings, although anonymous. Some of the buildings are leaning at ninety degrees; a few resemble old New York skyscrapers. All the buildings were made and sculpted with books. Liu said, "I was drawn to books at first because of the uniform density; the morphology of books seems to give them the ability to replace all other architectural and urban features; books represent a real-world and expand wantonly." [12] Liu used iron and hardware for the basic frame covered by wood to assemble the books into buildings.
Urs Fischer
Urs Fischer (1973-) was born in Switzerland; his parents were doctors. At first, Fisher studied photography before opening his studio. Today, his work includes sculpture, drawing, painting, and his first interest in photography. Fisher's work is frequently classified as subversive and non-traditional, often creating oversized installations. Get Up Girl a Sun Is Running the World (8.2.42) is an installation based on 2,000-year-old olive trees found in the Italian countryside. He cast the trees in aluminum and covered them with white enamel. The natural trees' gnarled trunk and twisted branches are captured and frozen in time, then installed in an abnormal setting of concrete and glass, the beauty of the natural setting lost. Fischer said, "What interests me about the 2000-year-old olive trees is the fact that once they are cast bare naked, they become Memoriam of condensed time. Through a cast olive tree, you can not only experience the lapse of real-time, that is lived time, frozen in its given form but through this transformation also a different calibrated temporality." [13]
Fischer invited 1,500 people to help make clay structures for his exhibition of Yes (8.2.43, 8.2.44). The clay was provided for the people, and they were able to use all the clay needed and create anything they wanted. The people were from all walks of life, no experience required, and the results from the people were varied; some sculptures were polished, others were crude; they were large and small, made by young and old. The exhibit covered all of the floor space, and some people even used the walls. Fischer worked beside the people making his sculptures, including the immense wax sculpture Untitled (8.2.45), an image resembling Giambologna's Rape of the Sabine Women from the 1500s. Fischer placed wicks throughout the wax sculpture and then burned them to achieve the dripping and melted look of the statue. His oversized sculpture contrasted the smaller creations of the people, and the exhibition has the look of disparate pieces in a landscape of ruins, broken figures, and unassociated images.
Fischer used the concept of ordinary objects to create the absurd. His monumental work, Untitled (Lamp/Bear) (8.2.46), is an image of a child's bright yellow teddy bear as part of a desk lamp. The mammoth bear is about two stories high as it sits, flopping forward. The bear appears to be sitting on the lamp as the light sits above his head, the giant lamp providing light for the outdoor space. The bear is wrinkled and looks cozy, ready to be held by a child. However, the bear is not made from soft fur as it appears; instead, it is cast of bronze and weighs about seventeen tons.
Olafur Eliasson
Olafur Eliasson (1967-) was born in Denmark, although his parents had recently emigrated from Iceland. His father was an artist and moved back to Iceland after divorcing Eliasson's mother. Eliasson studied art at the Royal Danish Academy of Fine Arts and then moved to Germany. His artwork is large and complex, so Eliasson collaborates with a team of architects and engineers, and other artists. He is also a professor at Berlin University, where he works with spatial experiments. Eliasson is well-known for creating multi-sensory exhibitions and combining scientific phenomena, environmental concerns, and art.
The weather has always been part of everyday conversation, will it be hot or cold or when the rain might start. Eliasson created The Weather Project (8.2.47) to bring the experience of the sun and sky. A fine mist is felt throughout the day as though coming from outside. At the end of the hall, hundreds of mono-frequency lamps, as installed in a semicircle, reflected in a huge mirror. The lamps contain a light restricted to a very narrow frequency, so they only emit yellow and black; any other colors become invisible. The result is a perfect sun seemingly static and unmoving, its light turning viewers into small black shadows. Eliasson has constructed the exhibition so viewers can also walk behind the sun and view how the electrical wiring, sub-structure, and misting machines are assembled and function. Aluminum frames were lined with the mirrored foil to create the reflective sun with the 200 lights in the semicircle.
The New York City Waterfalls (8.2.48) were a temporary installation during the summer of 2008 in New York City. The four waterfalls were situated on the East River at each bridge. The waterfalls were meant to enhance the public space and give people a sense of relationship with their surroundings. The basic structure was a set of scaffolding as the backbone for each separate fall, along with water pumps and hoses. The waterfalls were powered by electricity from "green power" and illuminated with LED lights. The fish were protected with special filters and intake pools to make the entire project environmentally sensitive. The cascading water brought a feeling of mountains and natural places in the middle of a congested urban area. The sound of falling water provides a softer sound among the harsh din of the city.
Eliasson wanted to bring the experience of the perspective of the horizon. He didn't think of the horizon as a line; instead, he felt it was a dimension. Eliasson explained, "Questioning one's horizon requires us to question linearity and create a new horizon." [15] His exhibit Inside the Horizon (8.2.49) is a series of forty-three triangular columns with varying widths running beside a grotto. Two sides of every column are covered with mirrors, and one side is clad in yellow glass tiles. The columns are illuminated from the inside. Each column is a different size and positioned at a different angle. Viewers walk through the columns, and their reflections create endless diffracted reflections on the water, building columns, and even themselves. The space becomes filled and changes with hundreds of facets and surprising visuals, much like a kaleidoscope.
Bharti Kher
Bharti Kher (1969-) was born in England and received her BA from Newcastle Polytechnic. She moved to India in 1993, where she still lives. Her art is based on a relationship of an object and its metaphysical and material dimensions and how she repositions the concepts and thoughts the viewer previously held. Most of her work includes the bindi, the red dot women in India apply to their forehead between the eyebrows. The red dot contains significant traditional and religious meanings and is generally associated with the Hindu definition of the third eye. Kher explained, "Many people believe it's a traditional symbol of marriage while others, in the West particularly, see it as a fashion accessory… But actually the bindi is meant to represent a third eye – one that forges a link between the real and the spiritual-conceptual worlds." [16] Kher uses the bindi as part of her art, shifting the meaning. Her sculptures are generally fantastical, blurring the lines between real and mythical, typically made from found objects with their own definition, which becomes changed. Kher makes her objects open to misinterpretation, magical thinking of abstract forms.
The Skin Speaks, a language not its own (8.2.50), is Kher's most well-known work. She became interested in the concept of elephants after seeing a photograph of a collapsed elephant being put into a truck, an image she remembered for a long time before making her elephant. The life-sized elephant is made from fiberglass and lies on the ground on its stomach with its head turned. If an elephant dies, it falls to the ground on its side, and this elephant is sited differently. The elephant is covered with white bindis forming unending patterns on the elephant. Kher used the bindi to act as a skin on the elephant giving life to something dying. The heavy elephant appears to be lifted by the thousands of bindis as the bindi do not stop while they move over the elephant's skin. The elephant's head is turned, and the eye appears to look at the viewer.
Kher wanted to create the heart of the massive blue sperm whale; however, she did not find enough scientific documentation about the construction of the whale's heart, so she invented her version in An Absence of Assignable Cause (8.2.51). Kher designed the life-sized heart and added veins and arteries jutting out from the enormous heart. The entire sculpture is covered with colored bindis, which Kher applies individually to create precise patterns and create movement. The monumental installation also reflects the position of the immense whale in the disintegrating environment.
Installation art is based on multiple earlier movements, including the reuse and small installations of Dada or Performance Art creatively using and interpreting space, even conceptual art and the focus on the ideas over the aesthetic.
[1] Retrieved from https://www.tate.org.uk/whats-on/tat...unflower-seeds
[2] Retrieved from https://www.magasin3.com/en/artwork/world-map-2/
[3] Retrieved from https://www.gap.net.au/yayoi-kusama
[4] Retrieved from https://mattress.org/works/infinity-...mirrored-room/
[5] Retrieved from https://www.tate.org.uk/art/artists/...-is-el-anatsui
[6] Retrieved from https://www.artnews.com/art-news/art...ungry-god-172/
[7] Retrieved from https://www.indiatoday.in/magazine/society-the-arts/story/20140127-subodh-gupta-and-his-creation-delhi-799864-1999-11-30
[8] Retrieved from https://www.npr.org/templates/story/...ryId=103674782
[9] Retrieved from https://stljewishlight.org/arts-ente...s-masterpiece/
[10] Retrieved from https://www.royalacademy.org.uk/arti...eginners-guide
[11] Retrieved by https://www.nytimes.com/2014/05/12/a...ino-plant.html
[12] Retrieved from http://whatsupmiami.blogspot.com/201...at-basels.html
[13] Retrieved from https://www.phillips.com/detail/ugo-...one/UK010411/4
[14] Retrieved from https://hyperallergic.com/69868/urs-...he-overcooked/
[15] Retrieved from https://www.fondationlouisvuitton.fr...de-the-horizon
[16] Retrieved from https://www.hauserwirth.com/artists/2793-bharti-kher/


1: Mathematics and Problem-Solving
We are surrounded by information—information in various forms such as numbers, charts, graphs, figures, sentences, songs, speeches, and conversations. Much of the information comes through media including television, magazines, and online communications. Many of the things we hear and read are true while many are not. How are we to determine what is true and what is not? What is truth anyway? Are there ways to detect false information? What are we to watch out for?
One of the primary goals of this book is to answer some of these questions through logic and mathematics. This is particularly important in the Internet-dominated, media-centered, and information-driven age in which we find ourselves living today. With new technologies, it is becoming much easier for businesses and organizations to send out deceptive messages and misleading advertisements, and they can target specific individuals based on their online activities. Throughout this book you will find data, numbers, and statements that can be presented honestly or dishonestly in a deceptive manner. They may be numbers, percentages, odds, statistics, or written sentences.
In this first chapter we study logic, both correct use of logic and logical fallacies. Before we get into the formal side of logic, however, we first look at some types of fallacies presented as English sentences. In other words, we will see some bad examples of logical arguments presented in common language. Then, in the following sections we will look carefully into some formal components of logic like propositions, truth tables, and logical equivalences.
Contributors and Attributions
-
Saburo Matsumoto
CC-BY-4.0


3.3: Thesis Statements
- Identify strong thesis statements
The thesis statment is the key to most academic writing. The purpose of academic writing is to offer your own insights, analyses, and ideas—to show not only that you understand the concepts you’re studying, but also that you have thought about those concepts in your own way and agreed or disagreed, or developed your own unique ideas as a result of your analysis. The thesis statement is the one sentence that encapsulates the result of your thinking, as it offers your main insight or argument in condensed form.
We often use the word “argument” in English courses, but we do not mean it in the traditional sense of a verbal fight with someone else. Instead, you “argue” by taking a position on an issue and supporting it with evidence. Because you’ve taken a position about your topic, someone else may be in a position to disagree (or argue) with the stance you have taken. Think about how a lawyer presents an argument or states their case in a courtroom—similarly, you want to build a case around the main idea of your essay. For example, in 1848, when Elizabeth Cady Stanton drafted “The Declaration of Sentiments,” she was thinking about how to convince New York State policymakers to change the laws to allow women to vote. Stanton was making an argument .
Some consider all writing a form of argument—or at least of persuasion. After all, even if you’re writing a letter or an informative essay, you’re implicitly trying to persuade your audience to care about what you’re saying. Your thesis statement represents the main idea—or point—about a topic or issue that you make in an argument. For example, let’s say that your topic is social media. A thesis statement about social media could look like one of the following sentences:
- Social media are hurting the communication skills of young Americans.
- Social media are useful tools for social movements.
A basic thesis sentence has two main parts:
- Topic: What you’re writing about
- Angle: What your main idea is about that topic
Thesis: A regular exercise regime leads to multiple benefits, both physical and emotional.
Topic: Regular exercise regime
Angle: Leads to multiple benefits
Thesis: Adult college students have different experiences than typical, younger college students.
Topic: Adult college students
Angle: Have different experiences
Thesis: The economics of television have made the viewing experience challenging for many viewers because shows are not offered regularly, similar programming occurs at the same time, and commercials are rampant.
Topic: Television viewing
Angle: Challenging because shows shifted, similar programming, and commercials
Identifying the Thesis Statement
You’ll remember that the first step of the reading process, previewing , allows you to get a big-picture view of the document you’re reading. This way, you can begin to understand the structure of the overall text. The most important step in getting a good understanding of an essay or book is to find the thesis statement.
A thesis consists of a specific topic and an angle on the topic. All of the other ideas in the text support and develop the thesis. The thesis statement is often found in the introduction, sometimes after an initial “hook” or interesting story; sometimes, however, the thesis is not explicitly stated until the end of an essay, and sometimes it is not stated at all. In those instances, there is an implied thesis statement, in which you can generally extract the thesis statement by looking for a few key sentences and ideas.
Most readers expect to see the point of your argument (the thesis statement) within the first few paragraphs. This does not mean that it has to be placed there every time. Some writers place it at the very end, slowly building up to it throughout their work, to explain a point after the fact. Others don’t bother with one at all but feel that their thesis is “implied” anyway. Beginning writers, however, should avoid the implied thesis unless certain of the audience. Almost every professor will expect to see a clearly discernible thesis sentence in the introduction.
Thesis statements vary based on the rhetorical strategy of the essay, but thesis statements typically share the following characteristics:
- Presents the main idea
- Most often is one sentence
- Tells the reader what to expect
- Is a summary of the essay topic
- Usually worded to have an argumentative edge
- Written in the third person
The following “How to Identify a Thesis Statement” video offers advice for locating a text’s thesis statement. It asks you to write one or two sentences that summarize the text. When you write that summary, without looking at the text itself, you’ve most likely paraphrased the thesis statement.
You can view the transcript for “How to Identify the Thesis Statement” here (download).
Writing a Thesis Statement
Remember your thesis should answer two simple questions: What issue are you writing about, and what is your position, or angle, on it?
A thesis statement is a single sentence (or sometimes two) that provides the answers to these questions clearly and concisely. Ask yourself, “What is my paper about, exactly?” to help you develop a precise and directed thesis, not only for your reader, but for you as well.
A good thesis statement will:
- Consist of just one idea
- Make your position clear
- Be specific
- Have evidence to support it
- Be interesting
- Be written clearly
A good basic structure for a thesis statement is “they say, I say.” What is the prevailing view, and how does your position differ from it? However, avoid limiting the scope of your writing with an either/or thesis under the assumption that your view must be strictly contrary to their view.
Following are some typical thesis statements:
- Although many readers believe Romeo and Juliet to be a tale about the ill fate of two star-crossed lovers, it can also be read as an allegory concerning a playwright and his audience.
- The “War on Drugs” has not only failed to reduce the frequency of drug-related crimes in America but actually enhanced the popular image of dope peddlers by romanticizing them as desperate rebels fighting for a cause.
- The bulk of modern copyright law was conceived in the age of commercial printing, long before the Internet made it so easy for the public to compose and distribute its own texts. Therefore, these laws should be reviewed and revised to better accommodate modern readers and writers.
- The usual moral justification for capital punishment is that it deters crime by frightening would-be criminals. However, the statistics tell a different story.
- If students really want to improve their writing, they must read often, practice writing, and receive quality feedback from their peers.
- Plato’s dialectical method has much to offer those engaged in online writing, which is far more conversational in nature than print.
Thesis Problems to Avoid
Although you have creative control over your thesis sentence, you still should try to avoid the following problems, not for stylistic reasons, but because they indicate a problem in the thinking that underlies the thesis sentence.
Practice identifying strong thesis statements in the following interactive.
https://lumenlearning.h5p.com/content/1290916673521810808/embed
argument : in writing, the argument is the main stance, claim, or position that is supported with evidence
explicit thesis : a clear and direct statement of the writer’s claim
thesis statement : a statement of the topic of the piece of writing and the angle the writer has on that topic
Contributors and Attributions
- Modification, adaptation, and original content. Provided by : Lumen Learning. License : CC BY: Attribution
- Annotating an Essay or Book. Provided by : Excelsior OWL. Located at : https://owl.excelsior.edu/orc/what-to-do-while-reading/annotating/annotating-an-essay-or-book/ . License : CC BY: Attribution
- Checklist for a Thesis Statement. Provided by : Excelsior OWL. Located at : https://owl.excelsior.edu/esl-wow/getting-ready-to-write/developing-a-thesis/esl-checklist-for-a-thesis-statement/ . License : CC BY: Attribution
- Judging Thesis Statements. Provided by : Excelsior OWL. Located at : https://owl.excelsior.edu/esl-wow/getting-ready-to-write/developing-a-thesis/esl-judging-thesis-statements/ . License : CC BY: Attribution
- Research Writing and Argument. Authored by : Pavel Zemliansky. Located at : learn.saylor.org/mod/page/view.php?id=7163. Project : Methods of Discovery: A Guide to Research Writing. License : CC BY: Attribution
- Parts of a Thesis Sentence and Common Problems. Provided by : Excelsior OWL. Located at : https://owl.excelsior.edu/writing-process/thesis-sentence/thesis-sentence-angles/ . License : CC BY: Attribution
- Argument in College Writing. Provided by : Excelsior OWL. Located at : https://owl.excelsior.edu/argument-and-critical-thinking/argument-in-college-writing/ . License : CC BY: Attribution
- How to Identify the Thesis Statement. Authored by : Martha Ann Kennedy. Located at : https://youtu.be/di1cQgc1akg . License : All Rights Reserved . License Terms : Standard YouTube License


1.7: Solving Problems Using Ratios
Learning Outcome
- Use ratios to solve simple word problems
When you apply for a mortgage, the loan officer will compare your total debt to your total income to decide if you qualify for the loan. This comparison is called the debt-to-income ratio. The ratio must fall below a certain threshold for a lender to agree to loan you money. You can improve your debt to income ratio either by increasing your income, decreasing your debt, or both.
Ratios
A ratio compares two numbers or two quantities and indicates their proportion to one another.
The ratio of \(a\) to \(b\) is written \(a\text{ to }b,{\Large\frac{a}{b}},\text{ or }\mathit{\text{a}}\text{ :}\mathit{\text{b}}\)
In this section, we will use the various notations interchangeably (i.e. \(\large\frac{5}{7}\) is the same as \(5\) : \(7\), is the same as \(5\) to \(7\)). When a ratio is written in fraction form, the fraction should be simplified. If it is an improper fraction, we do not change it to a mixed number. Because a ratio compares two quantities, we would leave a ratio as \({\large\frac{4}{1}}\) or \(4\) : \(1\) instead of simplifying it to \(4\) so that we can see the two parts of the ratio.
example
Write each ratio as a fraction:
- \(15\text{ to }27\)
- \(45\text{ to }18\)
Solution
| 1. | |
| \(\text{15 to 27}\) | |
| Write as a fraction with the first number in the numerator and the second in the denominator. | \({\Large\frac{15}{27}}\) |
| Simplify the fraction. | \({\Large\frac{5}{9}}\) |
| 2. | |
| \(\text{45 to 18}\) | |
| Write as a fraction with the first number in the numerator and the second in the denominator. | \({\Large\frac{45}{18}}\) |
| Simplify. | \({\Large\frac{5}{2}}\) |
Notice we leave the second ratio as an improper fraction.
try it
[ohm_question]146453[/ohm_question]
In the following video you will see more examples of how to express a ratio as a fraction.
Let’s consider some word problems with ratios. We will need to remember how to create equivalent ratios. As long as you are dividing or multiplying both parts of the ratio (or the numerator and denominator, if you are using a faction representation), then you are creating an equivalent ratio.
Take the ratio \(10\) : \(4\)
Let’s create an equivalent ratio by simplifying it. A common factor is \(2\), so let’s divide both values by \(2\). The result is a true statement of equivalent ratios.
\(10\) : \(4\) equals \(5\) : \(2\)
or
\(\frac{10}{4}\) = \(\frac{5}{2}\)
Let’s create another equivalent ratio by making it larger. Let’s multiply both values by \(3\). The result is a true statement of equivalent ratios.
\(10\) : \(4\) equals \(30\) : \(12\)
or
\(\frac{10}{4}\) = \(\frac{30}{12}\)
EXample
If Maria buys one dress for every three pairs of pants, and last year she bought nine pairs of pants, how many dresses did she buy last year?
Solution:
We will set up two equivalent ratio sentences like this:
_____ dress : _____ pants (Maria’s purchase ratio)
_____ dress : _____ pants (last year’s actual purchases)
Then we will fill them out with the values we know:
__\(\color{red}{1}\)__ dress : __\(\color{red}{3}\)__ pants
__\(\color{red}{?}\)__ dress : __\(\color{red}{9}\)__ pants
To make an equivalent ratio and to get from \(3\) pants to \(9\) pants, we know that we must have multiplied the initial value by \(3\). Let’s do the same to the dresses to complete the equivalent ratio.
\(3\cdot1=3\)
Therefore, if Maria bought nine pairs of pants last year, based on her purchase ratio, she also bought three dresses.
Try It
The ratio of Sophomores to Seniors at Mauve Hills High School is seven to ten. If there are 154 Sophomores enrolled this year, how many Seniors are there?
[reveal-answer q=”368717″]Show Answer[/reveal-answer]
[hidden-answer a=”368717″]
__\(\color{red}{7}\)__ Sophomores : __\(\color{red}{10}\)__ Seniors
__\(\color{red}{154}\)__ Sophomores : __\(\color{red}{?}\)__ Seniors
We know that to make equivalent fractions, the \(7\) must have been multiplied by something. So let’s divide \(154\) by \(7\) to find out what that multiplier was.
\(154\div{7}=22\)
Let’s take that multiplier of \(22\) and multiply it by the \(10\) Seniors in the original ratio to create an equivalent ratio.
\(10\cdot{22}=220\)
There are \(220\) Seniors at Mauve Hills High School.
[/hidden-answer]
The next examples go one step further and take a little more thought as to what value you are looking for and what to do with it after you calculate it.
ExAMPLE
The Pet Adoption Center in San Antonio houses a ratio of \(4\) cats to \(7\) dogs. If the shelter can house \(49\) dogs, how many total animals can they keep at the shelter?
[reveal-answer q=”587115″]Show Answer[/reveal-answer]
[hidden-answer a=”587115″]
First, we need to figure out how many cats are at the shelter.
\(\large{\frac{4}{7}}=\large{\frac{?}{49}}\)
\(\large{\frac{4}{7}}\cdot\color{red}{\frac{7}{7}}=\large{\frac{28}{49}}\)
There are \(28\) cats at the shelter. So in total there are \(28+49=77\) animals housed at the Pet Adoption Center in San Antonio.
[/hidden-answer]
TRY IT
[ohm_question]156943[/ohm_question]


Skip to main content
Study Questions
-
Identify the first four locations hooked up to the ARPANET
-
Describe the difference between the Internet and the World Wide Web
-
List three of your favorite Web 2.0 apps or websites
-
Identify the killer app for the Internet
-
List a few home internet connections
-
List a few business internet connections
-
Describe the difference between a LAN and a WAN
-
Describe the difference between an intranet and an extranet
-
Explain what a network topology is
-
Explain what powerline networking is
Exercises
-
Give an example of each of the following terms:
-
Wireless LAN (WLAN)
-
Wide-area network (WAN)
-
Intranet
-
Local-area network (LAN)
-
Extranet
-
Give an example for each of the following:
-
Fault tolerance
-
Scalability
-
Quality of service (QoS)
-
Security
-
Create a google account at -
google.com,
create a new document using google docs, share the document with others and explore document sharing via your google account.
-
Find the IP address of your computer. Explain the steps how you find it.
-
Identify your or your school’s Internet service provider.
-
Pretend that you are planning a trip to three foreign countries in the next month. Consult your wireless carrier to determine if your mobile phone would work properly in those countries. Identify if there are costs and other alternatives to have your phone work properly.


6.14: Summary
In this chapter we have formulated a complete methodology to derive the kinetics of systems of one or more point masses under the influence of forces and constraints generated by constraining objects and massless mechanisms. Here we have made use of the kinematics from the previous chapter, introduced the important concepts force and mass, and discussed the procedure of sketching the problem and the FBDs to obtain and solve the EoMs. Let us summarise the main topics and equations in this chapter:
- Mass and force
- Mass and force are defined by Newton’s laws
\(-\sum \overrightarrow{\boldsymbol{F}}=m \overrightarrow{\boldsymbol{a}}\)
\(-\overrightarrow{\boldsymbol{F}}_{\text {action }}=-\overrightarrow{\boldsymbol{F}}_{\text {reaction }}\)
- Procedure for solving the EoMs
- Sketch, CS, FBD, projecting forces, determining the EoM
- Solving the EoM using kinematics and constraints
- Relative motion and IRF
- Forces, mechanisms and constraints
- Gravity: \(\overrightarrow{\boldsymbol{F}}_{g}=m \overrightarrow{\boldsymbol{g}}\)
- Friction: Eqs. \((6.28,6.29)\)
- Massless mechanisms: \(\sum \overrightarrow{\boldsymbol{F}}_{\text {ext }}=\overrightarrow{\mathbf{0}}\)
- Rods and ropes: \(\left|\overrightarrow{\boldsymbol{r}}_{A}-\overrightarrow{\boldsymbol{r}}_{B}\right|=L\)
- Spring: \(\overrightarrow{\boldsymbol{F}}_{k}=k \Delta L \hat{\boldsymbol{s}}\)
- Damper: \(\overrightarrow{\boldsymbol{F}}_{c}=c \frac{\mathrm{d} L}{\mathrm{~d} t} \hat{\boldsymbol{s}}\)
Although all kinetic problems can be solved using the general method outlined in this chapter, certain problems, in particular those where the forces only depend on position, are easier to solve using the concepts of work and energy, which will be discussed in the next chapter. If the forces only depend on time, or have a very short duration, the principle of impulse and momentum is useful, which will be discussed in chapter 8 .


7.3: What are the roles of local, state and federal agencies in education?
-
- Last updated
- Save as PDF
By: Melissa Belliares
Learning Objectives
- Be able to differentiate educational roles of each level of the government
- Identify legislation and programs offered by the federal government...
- Be able to define the major roles the state play in the school system...
- Be able to understand the goals of the local government...
Federal
The federal government sets up departments, advisory committees and programs (such as the We the People Program) to ensure the state and local agencies follow the basic human rights for all citizens by implementing civil rights amendments and programs that endorse good citizenship. The departments also serve to allocate funds and ensure a good development of students and curriculum. Originally, the federal government's main use was land grants for states to develop colleges and universities. As world competition soared after and during the world wars of the 20th century, the U.S. government started to help with education assistance (
Wiki's GI Bill website
) for returning veterans; and then later, the National Defense department in 1958 set up allocations to help potential students better in all core areas like math and science (click here to find out more about the
National Science Foundation
). "In the State of the Union Address on January 31, 2006, United States President George W. Bush announced the American Competitiveness Initiative. Bush proposed the initiative to address shortfalls in federal government support of educational development and progress at all academic levels in the
STEM
fields" (STEM 2008).
The federal government also sets up a great resource tool for teachers. The Federal Resource for Educational Excellence (FREE) allows teachers to utilize the internet for classroom resources for all subjects and levels from the National Science Foundation and other federally funded programs like the Endowment for Humanities, National Park Service, Library of Congress, Smithsonian Institution, National Institutes of Health, and even the Labor Department among others.
Although the federal government has no constitutional role in education because of the 10th amendment , and no mention in the document pertaining to the federal role, in 2003, the Secretary of Education released the revision of the "constitutionally protected prayer" in public schools (originally from the Elementary and Secondary Act) and mandated by the No Child Left behind Act of 2001 (ASCD, 2003) (Findlaw 2008). The voucher system gives parents an option for their child's education through the faith-based initiatives issued by the white house in 2002 to "create a level playing field for faith and community based organizations to compete for USAID programs" (USAID). The executive branch has implemented National Charter Week for a week in May starting this year, 2008.
Giving options and opportunities to parents and students is the underlying theme in the federal's role in the educational system. They have the power to make changes and institute ideas that help students benefit more from the national educational system. The federal government's role in education also plays a socio-economic role on a national level because most of these changes dealt with funding and civil rights and a lot of controversy build awareness for some these issues.
State
Each state's department of education holds the certification role for teachers on a state level and the VEA (a non state-funded organization) also help teachers earn national certification. According to "Home Instruction in Virginia," guidelines set by the Department of Education of Virginia, home-school teachers must meet licensure requirement in addition to a high school diploma. Virginia's department of education has a great job resource site for state-certified teachers at Teach Virginia Education Career Center. Here , teachers can find job opportunities in state or out of state.
The following are offices that work under the Department of Education of Virginia for the improvement of all Virginia's schools.
Division of Student Assessment & School Improvement:
reporting of results for the commonwealth’s statewide assessment programs, develop the tests that make up the commonwealth’s statewide assessment system (like the SOL's), promotes student learning and achievement by assisting schools and school divisions in the implementation of effective instructional strategies and best practices and include school-level and division-level academic reviews, school improvement planning, and innovative programs such as the Partnership for Achieving Successful Schools (PASS).
Division of Finance:
makes payments to school divisions, including entitlement payments and state and federal reimbursement requests; collaborates with the Department of Planning and Budget and the General Assembly for budgets and provides assistance to public education funding to school divisions; oversees federal grant allocations; implements Department of Agriculture programs for lunch and breakfast with training and technical assistance to all schools
Division of Instruction:
works with each level to ensure instructional programs adhere to requirements; oversees gifted and physical (among elective others) education; oversees the SOL’s; implement early childhood education (i.e. Head Start)
Division of Policy & Communication:
oversees public communication between the citizens and the Board of Education; releases news and relays a summary news articles for members; help with members speeches
Division of Special Education & Student Services:
over sees the deaf and blind state schools; provides education, strategy and options for special education for parents and teachers; revises state special education requirements
Division of Teacher Education & Licensure:
implements state and federal regulations for licensure for preK-12 levels
Division of Technology & Career Education:
contains designated adult literacy office; promotes educational technology; calculates and reports state and federal accountability results
Local
Nobody can bring the school community together more than the collaboration of parents and teachers when it comes to general and special education. According to the guidelines set by the state, the superintendent collects curriculum information from home-school parents as well. Constitutionally, all students should have the same opportunities that can be implemented not only by the federal and state governments, but through local means as well. Together, the parents and the teachers with consideration of the students, can best represent the school's wants and needs. The local agencies have to have their submitted "programs" in order to get funding from either the state or the federal governments (under the ESEA of 1965, all programs must comply with all other programs under the act). There seems to be an overwhelming outcry for alternatives to public education by people who believe the local agencies are not doing enough for their schools; may it be for more home schools or school vouchers. But, the popular trend of privatizing various school programs has some parents wondering if this is a real alternative or just product placement. (Manning, 1999) The PTA (parent and teacher association) is locally and state funded with cooperation on the national level. It is the PTA that brings communities and schools together to ensure proper development of the student and school is being implemented. A good example of this is the Standards of Learning Test nationally mandated, state funded, locally implemented.
Exercise \(\PageIndex{1}\)
1. Which educational reforms are in chronological order ?
a. Gi Bill, Section 504, Higher Education Act, NCLB
b. Higher Education Act, National Defense Act, NCLB
c. NCLB, National Defense Act, Section 504
d. National Defense Act, Higher Education Act, IDEA
2. How does the National Park Service contribute to the educational system?
a. A teachers' classroom information tool; part of FREE
b. A program that helps teachers plant a tree in every school yard
c. A program that builds a playground on every school yard
d. Names parks after the most apreciated teachers
3. Mary’s' mom believes that parents should be involved as much as possible, which state funded organization implemented locally would her mom belong to?
a. ESEA
b. PTA
c. NCLB
d. VEA
4. Rapheal Carter said "The misconceptions about grammar posted on the 'Net are at least as bad as the misconceptions about evolution, presumably because American public schools do an equally bad job of teaching both.” Which of the following statements would he agree with most?
a. The federal government should mandate evolution to be taught in public schools
b. School's curriculum should be emphasized on grammar in English and and fair instruction on evolution in History and Science
c. Our teachers need better state funded training with the internet
d. Our local communities should donate more computers
- Answer
- 1. D
- 2. A
- 3. B
- 4. B
References
ASCD: Association for Supervision and Curriculum Development. (2003). ESEA/NCLB analysis. Retrieved September 5, 2008 from http://www.ascd.org/portal/site/ascd/menuitem.44cb9f9033aaf17cbfb3ffdb62108a0c/template.article?articleMgmtId=679d0f05c1520010VgnVCM1000003d01a8c0RCRD
FindLaw: U.S. Constitution (2008). The tenth amendment. Retrieved September 15, 2008 from http://caselaw.lp.findlaw.com/data/constitution/amendment10/
FREE: Federal Resources for Educational Excellence. (2008). Retrieved September 6, 2008 from www.free.ed.gov/index.cfm
Home Instruction In Virginia: Information for Parents 2008-2009. (2008). p5. Retrieved 15, 2008 from www.doe.virginia.gov/VDOE/Parents/homeinst.pdf
Manning, Steve. (1999). Students for sale. The Nation. Retrieved September 10, 2008 from www.thenation.com/doc/19990927/manning
Teach Virginia: Education Career Center (2008). Retrieved September 12, 2008 from http://www.teachvirginia.org/
STEM fields. (2008). Retrieved September 15, 2008 from en.Wikipedia.org/wiki/STEM_fields
U.S. Department of Education. (2008). The Federal Role In Education. Retrieved September 5, 2008 from http://www.ed.gov/about/overview/fed/role.html
USAID: FBCI- About the FBCI (2008). Retrieved September 12, 2008 from www.usaid.gov/our_work/global_partnerships/fbci/about.html
Virginia Department of Education: Division of teacher education and teacher licensure. (2008). Retrieved September 14, 2008 from www.doe.virginia.gov/VDOE/newvdoe/teached.html


12.6: Reliance
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Digital devices are inseparable from the new logistics and communications that are increasingly underpinning human activity. Devices come in a wide array of shapes and sizes and have an equally wide range of functions. Probably the most ubiquitous and familiar devices are personal computers and smartphones. For many people it is impossible to imagine life without the instant connectivity and wealth of information provided by the internet and accessed through such devices. Devices have thus become an integral, perhaps indispensable, part of human life. As these devices permeate society, it is conceivable that humans cede some of their humanity to the digital realm. Using the internet for many of our basic human functions, both individual and societal, effectively requires the internet to make up part of what it means to be human. In 1945 Vannevar Bush introduced his idea of a ‘memex’, which he described as a device in which an individual stores all his books, records, and communications, and which is mechanized so that it may be consulted with exceeding speed and flexibility. It is an enlarged intimate supplement to his memory. (Bush 1945)
Eerily prescient, Bush’s description accurately describes smartphones. The implication of this is that, thanks to such a device, the limited human mind can be freed up to perform the uniquely human capacities to imagine, associate and experiment.
Of course, such reliance on technology can have negative consequences. If the technology was to disappear or be denied to us, we could potentially lose some of our humanity. The example of Egypt’s internet services being cut off demonstrates the large-scale vulnerability of the technology, as do the cyberattacks on Estonia in 2007 that lost citizens access to essential services such as banking. Consider Facebook, a social networking platform with over one billion users. Facebook, and its subsidiary Instagram, are used today as photograph repositories. Hundreds of millions of people upload photos as they are taken, effectively replacing the physical photo albums that older generations typically kept in their homes. Facebook thereby becomes an archive of visual memories. If the internet malfunctioned, Facebook, and the memories it contains, would be inaccessible.  The example of memories shows how over-reliance on technology for important human functions may be unwise.


7.2: Jurisdiction
jurisdiction . Jurisdiction refers to the legal authority to hear and decide a case (legal suit).
Jurisdiction Based on the Function of the Court
Trial Courts versus Appellate Courts
Jurisdiction Based on Subject Matter
“The [subject matter] jurisdictional distinction . . . tends to be utilized primarily in distinguishing between different trial courts. Appellate courts ordinarily can hear all types of cases, although there are several states that have separate appellate courts for criminal and civil appeals. At the trial level, most states have established one or more specialized courts to deal with particular legal fields. The most common areas delegated to specialized courts are wills and estates (assigned to courts commonly known as probate . . . courts), divorce, adoption or other aspects of family law (family or domestic relations courts), and actions based on the English law of equity (chancery courts). The federal system also includes specialized courts for such areas as customs and patents. While significant, the specialized courts represent only a small portion of all trial courts. Most trial courts are not limited to a particular subject but may deal with all fields. Such trial courts are commonly described as having general jurisdiction since they cover the general (i.e., non-specialized) areas of law. Criminal cases traditionally are assigned to courts with general jurisdiction.” [1]
Jurisdiction Based on the Seriousness of the Case
courts of limited jurisdiction only have authority to try infractions, violations, and petty crimes (misdemeanors) whereas other trial courts, called courts of general jurisdiction , have authority to try serious crimes (felonies) as well as minor crimes and offenses.
Jurisdiction Based on the Court’s Authority over the Parties to the Case
Jurisdiction based on State and Federal Autonomy (Geography)
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Kerper, H. B. (1979).
Introduction to the criminal justice system
(2nd ed.). West Publishing Company.
↵


6.4: Solve Mixture Applications with Systems of Equations
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Learning Objectives
By the end of this section, you will be able to:
- Solve mixture applications
- Solve interest applications
- Solve applications of cost and revenue functions
Before you get started, take this readiness quiz.
Solve Mixture Applications
Mixture application involve combining two or more quantities. When we solved mixture applications with coins and tickets earlier, we started by creating a table so we could organize the information. For a coin example with nickels and dimes, the table looked like this:
Using one variable meant that we had to relate the number of nickels and the number of dimes. We had to decide if we were going to let n be the number of nickels and then write the number of dimes in terms of n , or if we would let d be the number of dimes and write the number of nickels in terms of d .
Now that we know how to solve systems of equations with two variables, we’ll just let n be the number of nickels and d be the number of dimes. We’ll write one equation based on the total value column, like we did before, and the other equation will come from the number column.
For the first example, we’ll do a ticket problem where the ticket prices are in whole dollars, so we won’t need to use decimals just yet.
Example \(\PageIndex{1}\)
Translate to a system of equations and solve:
A science center sold 1,363 tickets on a busy weekend. The receipts totaled $12,146. How many $12 adult tickets and how many $7 child tickets were sold?
- Answer
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Step 1. Read the problem. We will create a table to organize the information. Step 2. Identify what we are looking for. We are looking for the number of adult tickets
and the number of child tickets sold.Step 3. Name what we are looking for. Let \(a= \text{the number of adult tickets.}\)
\(c= \text{the number of child tickets}\)A table will help us organize the data.
We have two types of tickets, adult and child.Write in a and c for the number of tickets. Write the total number of tickets sold at the bottom
of the Number column.Altogether 1,363 were sold. Write the value of each type of ticket in the
Value column.The value of each adult ticket is $12.
The value of each child tickets is $7.The number times the value gives the total value,
so the total value of adult tickets is \(a·12=12a\),
and the total value of child tickets is \(c·7=7c\).Fill in the Total Value column. Altogether the total value of the tickets was $12,146. Step 4. Translate into a system of equations. The Number column and the Total value column
give us the system of equations.We will use the elimination method to solve
this system. Multiply the first equation by \(−7\).Simplify and add, then solve for a . Substitute \(a=521\) into the first equation, then
solve for c .Step 6. Check the answer in the
problem.
521 adult at $12 per ticket makes $ 6,252
842 child at $7 per ticket makes $58,994
The total receipts are $12,146\(\checkmark\)Step 7. Answer the question. The science center sold 521 adult tickets and
842 child tickets.
Example \(\PageIndex{2}\)
Translate to a system of equations and solve:
The ticket office at the zoo sold 553 tickets one day. The receipts totaled $3,936. How many $9 adult tickets and how many $6 child tickets were sold?
- Answer
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206 adults, 347 children
Example \(\PageIndex{3}\)
Translate to a system of equations and solve:
The box office at a movie theater sold 147 tickets for the evening show, and receipts totaled $1,302. How many $11 adult and how many $8 child tickets were sold?
- Answer
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42 adults, 105 children
In the next example, we’ll solve a coin problem. Now that we know how to work with systems of two variables, naming the variables in the ‘number’ column will be easy.
Example \(\PageIndex{4}\)
Translate to a system of equations and solve:
Juan has a pocketful of nickels and dimes. The total value of the coins is $8.10. The number of dimes is 9 less than twice the number of nickels. How many nickels and how many dimes does Juan have?
- Answer
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Step 1. Read the problem.
We will create a table to organize the information.Step 2. Identify what we are looking for. We are looking for the number of
nickels and the number of dimes.Step 3. Name what we are looking for. Let \(n= \text{the number of nickels.}\)
\(d= \text{the number of dimes}\)A table will help us organize the data.
We have two types of coins, nickels and dimes.Write n and d for the number of
each type of coin.Fill in the Value column with the value of each
type of coin.The value of each nickel is $0.05.
The value of each dime is $0.10.The number times the value gives the total
value, so, the total value of the nickels is
\(n(0.05)=0.05n\) and the total value of dimes is
\(d(0.10)=0.10d\).
Altogether the total value of the coins is $8.10.Step 4. Translate into a system of equations. The Total Value column gives one equation. We also know the number of dimes is 9 less than
twice the number of nickels.Translate to get the second equation. Now we have the system to solve. Step 5. Solve the system of equations
We will use the substitution method.Substitute \(d=2n−9\) into the first equation. Simplify and solve for n . To find the number of dimes, substitute
\(n=36\) into the second equation.Step 6. Check the answer in the problem
63 dimes at \($0.10=$6.30\)
36 nickels at \($0.05=$1.80\)
Total \(=$8.10\checkmark\)Step 7. Answer the question. Juan has 36 nickels and 63 dimes.
Example \(\PageIndex{5}\)
Translate to a system of equations and solve:
Matilda has a handful of quarters and dimes, with a total value of $8.55. The number of quarters is 3 more than twice the number of dimes. How many dimes and how many quarters does she have?
- Answer
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13 dimes and 29 quarters
Example \(\PageIndex{6}\)
Translate to a system of equations and solve:
Priam has a collection of nickels and quarters, with a total value of $7.30. The number of nickels is six less than three times the number of quarters. How many nickels and how many quarters does he have?
- Answer
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19 quarters and 51 nickels
Some mixture applications involve combining foods or drinks. Example situations might include combining raisins and nuts to make a trail mix or using two types of coffee beans to make a blend.
Example \(\PageIndex{7}\)
Translate to a system of equations and solve:
Carson wants to make 20 pounds of trail mix using nuts and chocolate chips. His budget requires that the trail mix costs him $7.60. per pound. Nuts cost $9.00 per pound and chocolate chips cost $2.00 per pound. How many pounds of nuts and how many pounds of chocolate chips should he use?
- Answer
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Step 1. Read the problem.
We will create a table to organize the information.Step 2. Identify what we are looking for. We are looking for the number of pounds of
nuts and the number of pounds of chocolate
chips.Step 3. Name what we are looking for. Let \(n= \text{the number of pound of nuts.}\)
\(c= \text{the number of pounds of chips}\)Carson will mix nuts and chocolate chips to get
trail mix.
Write in n and c for the number of pounds of
nuts and chocolate chips.There will be 20 pounds of trail mix.
Put the price per pound of each item in
the Value column.
Fill in the last column using
\(\text{Number}•\text{Value}=\text{Total Value}\)Step 4. Translate into a system of equations.
We get the equations from the Number
and Total Value columns.Step 5. Solve the system of equations
We will use elimination to solve the system.
Multiply the first equation by \(−2\) to eliminate c.Simplify and add.
Solve for n .To find the number of pounds of chocolate
chips, substitute \(n=16\) into the first equation,
then solve for c .Step 6. Check the answer in the problem.
\(\begin{array} {lll} 16+4 &= &20\checkmark \\ 9·16+2·4 &= &152\checkmark \end{array}\)Step 7. Answer the question. Carson should mix 16 pounds of nuts with 4
pounds of chocolate chips to create the trail
mix.
Example \(\PageIndex{8}\)
Translate to a system of equations and solve:
Greta wants to make 5 pounds of a nut mix using peanuts and cashews. Her budget requires the mixture to cost her $6 per pound. Peanuts are $4 per pound and cashews are $9 per pound. How many pounds of peanuts and how many pounds of cashews should she use?
- Answer
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3 pounds peanuts and 2 pounds cashews
Example \(\PageIndex{9}\)
Translate to a system of equations and solve:
Sammy has most of the ingredients he needs to make a large batch of chili. The only items he lacks are beans and ground beef. He needs a total of 20 pounds combined of beans and ground beef and has a budget of $3 per pound. The price of beans is $1 per pound and the price of ground beef is $5 per pound. How many pounds of beans and how many pounds of ground beef should he purchase?
- Answer
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10 pounds of beans, 10 pounds of ground beef
Another application of mixture problems relates to concentrated cleaning supplies, other chemicals, and mixed drinks. The concentration is given as a percent. For example, a 20% concentrated household cleanser means that 20% of the total amount is cleanser, and the rest is water. To make 35 ounces of a 20% concentration, you mix 7 ounces (20% of 35) of the cleanser with 28 ounces of water.
For these kinds of mixture problems, we’ll use “percent” instead of “value” for one of the columns in our table.
Example \(\PageIndex{10}\)
Translate to a system of equations and solve:
Sasheena is lab assistant at her community college. She needs to make 200 milliliters of a 40% solution of sulfuric acid for a lab experiment. The lab has only 25% and 50% solutions in the storeroom. How much should she mix of the 25% and the 50% solutions to make the 40% solution?
- Answer
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Step 1. Read the problem.
A figure may help us visualize the
situation, then we will create a table to
organize the information.Sasheena must mix some of the \(25%\) solution and
some of the \(50%\) solution together to get \(200\space ml\) of
the \(40%\) solution.Step 2. Identify what we are looking for. We are looking for how much of each solution she
needs.Step 3. Name what we are looking for. Let \(x= \text{number of }ml\text{ of }25% \text{ solution.}\)
\(y= \text{number of }ml\text{ of }50%\text{ solution\)A table will help us organize the data. She will
mix x \(ml\) of \(25%\) with y \(ml\) of \(50%\) to get \(200 \space ml\)
of \(40%\) solution. We write the percents as decimals
in the chart.
We multiply the number of units times the
concentration to get the total amount of
sulfuric acid in each solution.Step 4. Translate into a system of
equations.
We get the equations from the Number
column and the Amount column.
Now we have the system.Step 5. Solve the system of equations
We will solve the system by elimination.
Multiply the first equation by \(−0.5\) to
eliminate y .Simplify and add to solve for x . To solve for y , substitute \(x=80\) into the first
equation.Step 6. Check the answer in the problem.
\(\begin{array} {lll} 80+120 &= &200\checkmark \\ 0.25(80)+0.50(120) &= &200\checkmark \\ {} &{} &\text{Yes!} \end{array} \)Step 7. Answer the question. Sasheena should mix \(80 \space ml\) of the \(25%\) solution with
\(120 \space ml\) of the \(50%\) solution to get the \(200\space ml\) of the
\(40%\) solution.
Example \(\PageIndex{11}\)
Translate to a system of equations and solve:
LeBron needs 150 milliliters of a 30% solution of sulfuric acid for a lab experiment but only has access to a 25% and a 50% solution. How much of the 25% and how much of the 50% solution should he mix to make the 30% solution?
- Answer
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120 ml of 25% solution and 30 ml of 50% solution
Example \(\PageIndex{12}\)
Translate to a system of equations and solve:
Anatole needs to make 250 milliliters of a 25% solution of hydrochloric acid for a lab experiment. The lab only has a 10% solution and a 40% solution in the storeroom. How much of the 10% and how much of the 40% solutions should he mix to make the 25% solution?
- Answer
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125 ml of 10% solution and 125 ml of 40% solution
Solve Interest Applications
The formula to model simple interest applications is \(I=Prt\). Interest, I , is the product of the principal, P , the rate, r , and the time, t . In our work here, we will calculate the interest earned in one year, so t will be 1.
We modify the column titles in the mixture table to show the formula for interest, as you’ll see in the next example.
Translate to a system of equations and solve:
Adnan has $40,000 to invest and hopes to earn \(7.1%\) interest per year. He will put some of the money into a stock fund that earns 8% per year and the rest into bonds that earns 3% per year. How much money should he put into each fund?
- Answer
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Did you notice that the Principal column represents the total amount of money invested while the Interest column represents only the interest earned? Likewise, the first equation in our system, \(s+b=40,000\), represents the total amount of money invested and the second equation, \(0.08s+0.03b=0.071(40,000)\), represents the interest earned.Step 1. Read the problem. A chart will help us organize the information. Step 2. Identify what we are looking for. We are looking for the amount to invest in each fund. Step 3. Name what we are looking for. Let \(s= \text{the amount invested in stocks.}\)
\(b= \text{the amount invested in stocks}\)Write the interest rate as a decimal for
each fund.
Multiply: Principal · Rate · TimeStep 4. Translate into a system of
equations.
We get our system of equations from
the Principal column and the
Interest column.Step 5. Solve the system of equations
by elimination.
Multiply the top equation by \(−0.03\).Simplify and add to solve for s . To find b , substitute s = 32,800 into
the first equation.Step 6. Check the answer in the
problem.We leave the check to you. Step 7. Answer the question. Adnan should invest $32,800 in stock and
$7,200 in bonds.
Example \(\PageIndex{14}\)
Translate to a system of equations and solve:
Leon had $50,000 to invest and hopes to earn \(6.2%\) interest per year. He will put some of the money into a stock fund that earns 7% per year and the rest in to a savings account that earns 2% per year. How much money should he put into each fund?
- Answer
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$42,000 in the stock fund and $8000 in the savings account
Example \(\PageIndex{15}\)
Translate to a system of equations and solve:
Julius invested $7000 into two stock investments. One stock paid 11% interest and the other stock paid 13% interest. He earned \(12.5%\) interest on the total investment. How much money did he put in each stock?
- Answer
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$1750 at 11% and $5250 at 13%
The next example requires that we find the principal given the amount of interest earned.
Example \(\PageIndex{16}\)
Translate to a system of equations and solve:
Rosie owes $21,540 on her two student loans. The interest rate on her bank loan is \(10.5%\) and the interest rate on the federal loan is \(5.9%\). The total amount of interest she paid last year was \($1,669.68\). What was the principal for each loan?
- Answer
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Step 1. Read the problem. A chart will help us organize the information. Step 2. Identify what we are looking for. We are looking for the principal of each loan. Step 3. Name what we are looking for. Let \(b= \text{the principal for the bank loan.}\)
\(f= \text{the principal on the federal loan}\)The total loans are $21,540. Record the interest rates as decimals
in the chart.
Multiply using the formula I = Prt to
get the Interest.Step 4. Translate into a system of
equations.
The system of equations comes from
the Principal column and the Interest
column.Step 5. Solve the system of equations
We will use substitution to solve.
Solve the first equation for b .Substitute b = − f + 21.540 into
the second equation.Simplify and solve for f . To find b , substitute f = 12,870 into the first equation. Step 6. Check the answer in the
problem.We leave the check to you. Step 7. Answer the question. The principal of the federal loan was $12,870 and
the principal for the bank loan was $8,670.
Example \(\PageIndex{17}\)
Translate to a system of equations and solve:
Laura owes $18,000 on her student loans. The interest rate on the bank loan is 2.5% and the interest rate on the federal loan is 6.9%. The total amount of interest she paid last year was $1,066. What was the principal for each loan?
- Answer
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Bank $4,000; Federal $14,000
Example \(\PageIndex{18}\)
Translate to a system of equations and solve:
Jill’s Sandwich Shoppe owes $65,200 on two business loans, one at 4.5% interest and the other at 7.2% interest. The total amount of interest owed last year was $3,582. What was the principal for each loan?
- Answer
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$41,200 at 4.5%, $24,000 at 7.2%
Solve applications of cost and revenue functions
Suppose a company makes and sells x units of a product. The cost to the company is the total costs to produce x units. This is the cost to manufacture for each unit times x , the number of units manufactured, plus the fixed costs.
The revenue is the money the company brings in as a result of selling x units. This is the selling price of each unit times the number of units sold.
When the costs equal the revenue we say the business has reached the break-even point .
COST AND REVENUE FUNCTIONS
The cost function is the cost to manufacture each unit times x , the number of units manufactured, plus the fixed costs.
\[C(x)=(\text{cost per unit})·x+\text{fixed costs}\nonumber \]
The revenue function is the selling price of each unit times x , the number of units sold.
\[R(x)=(\text{selling price per unit})·x\nonumber \]
The break-even point is when the revenue equals the costs.
\[C(x)=R(x)\nonumber\]
Example \(\PageIndex{19}\)
The manufacturer of a weight training bench spends $105 to build each bench and sells them for $245. The manufacturer also has fixed costs each month of $7,000.
ⓐ Find the cost function C when x benches are manufactured.
ⓑ Find the revenue function R when x benches are sold.
ⓒ Show the break-even point by graphing both the Revenue and Cost functions on the same grid.
ⓓ Find the break-even point. Interpret what the break-even point means.
- Answer
-
ⓐ The manufacturer has $7,000 of fixed costs no matter how many weight training benches it produces. In addition to the fixed costs, the manufacturer also spends $105 to produce each bench. Suppose x benches are sold.
\(\begin{array} {ll} {\text{Write the general Cost function formula.}} &{C(x)=(\text{cost per unit})·x+\text{fixed costs}} \\ {\text{Substitute in the cost values.}} &{C(x)=105x+7000} \\ \end{array}\)
ⓑ The manufacturer sells each weight training bench for $245. We get the total revenue by multiplying the revenue per unit times the number of units sold.
\(\begin{array} {ll} {\text{Write the general Revenue function.}} &{C(x)=(\text{selling price per unit})·x} \\ {\text{Substitute in the revenue per unit.}} &{R(x)=245x} \\ \end{array}\)
ⓒ Essentially we have a system of linear equations. We will show the graph of the system as this helps make the idea of a break-even point more visual.
\[\left\{ \begin{array} {l} C(x)=105x+7000 \\ R(x)=245x \end{array} \right. \quad \text{or} \quad \left\{ \begin{array} {l} y=105x+7000 \\ y=245x \end{array} \right. \nonumber \]
ⓓ To find the actual value, we remember the break-even point occurs when costs equal revenue.
\(\begin{array} {ll} {\text{Write the break-even formula.}} &{\begin{array} {l} {C(x)=R(x)} \\ {105x+7000=245x} \end{array}} \\ {\text{Solve.}} &{\begin{array} {l} {7000=140x} \\ {50=x} \end{array}} \\ \end{array}\)
When 50 benches are sold, the costs equal the revenue.
When 50 benches are sold, the revenue and costs are both $12,250. Notice this corresponds to the ordered pair \((50,12250)\).
Example \(\PageIndex{20}\)
The manufacturer of a weight training bench spends $15 to build each bench and sells them for $32. The manufacturer also has fixed costs each month of $25,500.
ⓐ Find the cost function C when x benches are manufactured.
ⓑ Find the revenue function R when x benches are sold.
ⓒ Show the break-even point by graphing both the Revenue and Cost functions on the same grid.
ⓓ Find the break-even point. Interpret what the break-even point means.
- Answer
-
ⓐ \(C(x)=15x+25,500\)
ⓑ \(R(x)=32x\)
ⓒ
ⓓ 1,5001,500; when 1,500 benches are sold, the cost and revenue will be both 48,000
Example \(\PageIndex{21}\)
The manufacturer of a weight training bench spends $120 to build each bench and sells them for $170. The manufacturer also has fixed costs each month of $150,000.
ⓐ Find the cost function C when x benches are manufactured.
ⓑ Find the revenue function R when x benches are sold.
ⓒ Show the break-even point by graphing both the Revenue and Cost functions on the same grid.
ⓓ Find the break-even point. Interpret what the break-even point means.
- Answer
-
ⓐ \(C(x)=120x+150,000\)
ⓑ \(R(x)=170x\)
ⓒ
ⓓ \(3,000\); when 3,000 benches are sold, the revenue and costs are both $510,000
Access this online resource for additional instruction and practice with interest and mixtures.
- Interest and Mixtures
Key Concepts
-
Cost function:
The cost function is the cost to manufacture each unit times
x
, the number of units manufactured, plus the fixed costs.
\(C(x)=(\text{cost per unit})·x+\text{fixed costs}\)
-
Revenue:
The revenue function is the selling price of each unit times
x
, the number of units sold.
\(R(x)=(\text{selling price per unit})·x\)
-
Break-even point:
The break-even point is when the revenue equals the costs.
\(C(x)=R(x)\)
Glossary
- cost function
- The cost function is the cost to manufacture each unit times xx, the number of units manufactured, plus the fixed costs; C ( x ) = (cost per unit) x + fixed costs.
- revenue
- The revenue is the selling price of each unit times x , the number of units sold; R(x ) = (selling price per unit) x .
- break-even point
- The point at which the revenue equals the costs is the break-even point; C(x)=R(x).