To solve this problem, I need to find the relationship between the train's position on the trestle and its height above the gorge. I can do this by finding the equation of the line that passes through the endpoints of the trestle, which are (0, 172) and (80, 112). To find the equation of a line, I need its slope and its y-intercept. The slope is the change in y over the change in x, which is $\frac{112 - 172}{80 - 0} = -\frac{60}{80} = -\frac{3}{4}$. The y-intercept is the value of y when x is zero, which is 172. So the equation of the line is $y = -\frac{3}{4}x + 172$. This means that for any x-coordinate on the trestle, I can plug it into this equation and get the corresponding y-coordinate, which is the height above the gorge. Now I need to find the x-coordinate of the front of the train when it has traveled $\frac{3}{4}$ of the trestle's length. The trestle's length is 80 feet, so $\frac{3}{4}$ of that is 60 feet. This means that the front of the train is at the point where x is 60. To find the height of the train at this point, I need to plug x = 60 into the equation of the line and get the value of y. So $y = -\frac{3}{4}(60) + 172 = -45 + 172 = 127$. This means that the front of the train is 127 feet above the bottom of the gorge when it has traveled $\frac{3}{4}$ of the trestle's length.


A railroad trestle spans a gorge 80 feet wide and connects two cliffs at heights of 112 and 172 feet above the bottom of the gorge. A train is crossing this gorge from the higher cliff to the lower. When the front of the train has traveled $\frac{3}{4}$ of the trestle's length, how many feet is it above the bottom of the gorge?