College Algebra, Chapter 4, 4.4, Section 4.4, Problem 100

Suppose that an open box with a volume of $1500\text{cm}^3$ is to be constructed by taking a
piece of cardboard $20$cm by $40$cm, cutting squares of side length $x$ can from each corner, and folding up the sides.
Two different ways, and find the exact dimensions of the box in each case.



If the volume of the box is $1500\text{cm}^3$, then

$
\begin{equation}
\begin{aligned}
x(40-2x)(20-2x) &= 1500 && \text{Model}\\
\\
(40 -2x^2)(20-2x) &= 1500 && \text{Distribute } x\\
\\
800x - 80x^2 - 40x^2 + 4x^3 &= 1500 && \text{Apply FOIL method}\\
\\
800x - 120x^2 + 4x^3 &= 1500 && \text{Combine like terms}\\
\\
200x - 30x^2 + x^3 &= 375 && \text{Divide both sides by 4}\\
\\
x^3 - 30x^2 + 200x - 375 &= 0
\end{aligned}
\end{equation}
$

The, by using synthetic division and trial and error with the factor of 375,


Thus,

$
\begin{equation}
\begin{aligned}
x^3 - 30x^2 + 200x - 375 & = 0 \\
\\
(x-5)(x^2 -25x + 75) &= 0
\end{aligned}
\end{equation}
$

So if $x = 5$, then

$
\begin{equation}
\begin{aligned}
40 - 2x &= 30 \\
\\
20 - 2x &= 10
\end{aligned}
\end{equation}
$

Therefore, the exact dimension of the box is $5$ by $30$ by $10$cm.