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.