<s>[INST] Volume of a Rectangular Prism when given only the Surface Area So, in my equation, they ask you to find the volume of a rectangular prism when you only are given the surface area. As an example, one of the equations gave you a surface area of 240 yards squared, and no side measurements of any sort. How can I solve this if supposedly you need the length, width and height to find the volume?
Thank you for your time.
 [/INST] As the other answer suggests, if we don't place any restriction on
what values the sides of a cuboid can take, there are way too many possibilities.
We will assume the sides of the cuboid are all integers.
Given a cuboid of dimension $a \times b \times c$ where $a,b,c$ are integers, 
we have
$$\begin{align}
\verb/Area/ &= 2(ab+bc+ca)\\
\verb/Vol/  &= abc\\
\end{align}$$
WOLOG, we will assume $a \le b \le c$ and rewrite the formula for area as
$$(a+b)(a+c) = (ab+bc+ca)+a^2 = \frac12\verb/Area/+a^2 = 120+a^2$$
Since $$\verb/Area/ = 240 \implies 120 = (ab+bc+ca) \ge 3a^2 \implies
1 \le a \le 6$$
We have only $6$ cases to analysis.
$$\begin{array}{clll}
a = 1 &\implies 120+a^2 = 121 = 11\times 11 &\leadsto& (a,b,c,\verb/Vol/) = (1,10,10,100)\\
a = 2 &\implies 120+a^2 = 124 = 4 \times 31 &\leadsto& (a,b,c,\verb/Vol/) = (2,2, 29,116)\\
a = 3 &\implies 120+a^2 = 129 = 3 \times 43 &\leadsto& \verb/nothing/\\
a = 4 &\implies 120+a^2 = 136 = 2^3 \times 17 &\leadsto& (a,b,c,\verb/Vol/) = (4,4,13,208)\\
a = 5 &\implies 120+a^2 = 145 = 5 \times 29   &\leadsto& \verb/nothing/\\
a = 6 &\implies 120+a^2 = 156 = 2^2 \times 3 \times 13 &\leadsto&
 (a,b,c,\verb/Vol/) = (6,6,7,252)
\end{array}$$
To summarize, if the sides of the cuboids are all integers, there are 4 possible
volumes $100, 116, 208, 252$.
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