Question

Determine the dimensions of a rectangular solid (with square base) with a maximum volume if its surface are is 337.5 square centimeters.

Answer 1

surface area**

Answer 2

\[Volume=x^2y\\
Surface~area=2x^2+4xy\]
You know that \(2x^2+4xy=337.5\), and you want to maximize volume. Express volume in as a function of one variable only; to do this, use the surface area equation:
\[337.5=2x^2+4xy~\Rightarrow~\color{blue}{y=\frac{337.5-2x^2}{4x}}\]
Substitute the blue expression into the volume equation:
\[V=x^2\left(\frac{337.5-2x^2}{4x}\right)\\
V=\frac{1}{4}\left(337.5x-2x^3\right)\]
To maximize \(V\), find \(V'\) and determine the intervals on which \(V\) is increasing/decreasing. In other words, use the first derivative test to find the critical points of \(V\):
\[V'=\frac{1}{4}\left(337.5-6x^2\right)\]
Critical point(s) is/are
\[\frac{1}{4}\left(337.5-6x^2\right)=0\\
x^2=56.25\\
x=\sqrt{56.25}\]
(Ignore the negative root. This is because you're dealing with dimensions of a box, which must be greater than zero.)

\(x\) is confined to the interval \(0<x<\infty\), so when you're finding where \(V\) is increasing/decreasing, you'll be using the intervals \(0<x<\sqrt{56.25}\) and \(\sqrt{56.25}<x<\infty\).