College Algebra, Chapter 4, 4.2, Section 4.2, Problem 84

Suppose that a cardboard box has a square base, with each edge of the base has length $x$ inches. The total length of all 12 edges of the box is 144in.
a.) Show that the volume of the box is given by the function $V(x) = 2x^2 ( 18 - x)$
b.) What is the domain of $V$? (Use the fact that length and volume must be positive.)
c.) Draw a graph of the function $V$ and use it to estimate the maximum volume for such a box.


a.) Recall that the formula for the perimeter of the box with square base with edge $x$ is $P = 8x + 4y$. Solving for $y$, we have

$
\begin{equation}
\begin{aligned}
144 &= 8x + 4y && \text{Divide both sides by 4} \\
\\
36 &= 2x + y\\
\\
y &= 36 - 2x
\end{aligned}
\end{equation}
$

Recall that the volume of the box is $V = x^2 y$

$
\begin{equation}
\begin{aligned}
V &= x^2 (36 - 2x)\\
\\
V &= 2x^2 (18 -x)
\end{aligned}
\end{equation}
$


b.) If $V$ can never be a negative value, then the domain of $V$ is $(-\infty, 18]$
c.)


Based from the graph, the maximum volume is approximately $1730 \text{ in}^3$