Single Variable Calculus, Chapter 1, 1.1, Section 1.1, Problem 57

A rectangular piece of cardboard is used to construct a box with an open top having a dimensions 12 inches by 20 inches, by cutting out equal squares of side $x$ at each corner and then folding up the sides as shown in the figure. Express the volume $V$ of the box as a function of $x$






By thorough investigation, the dimensions of the open top box could be...






The volume of the box is equal to the product of its length, width and height. By using the dimension of the box, the volume we obtain is...



$
\begin{equation}
\begin{aligned}

\rm{Volume} &= x(20-2x)(12-2x) && (\text{Distribute } x \text{ in the equation})\\
\rm{Volume} & = (20x-2x^2)(12-2x) && (\text{Using FOIL method.})\\
\rm{Volume} & = 240x - 40x^2-24x^2 +4x^3 && (\text{Combine like terms.})\\
\end{aligned}
\end{equation}
$



$
\fbox{$\rm{Volume} = 240x - 64x^2+4x^3$}
$