## 2070.33 – A Rectangular Box

A rectangular box without top is formed by cutting squares from the corners of an 8 x 15 sheet of cardboard and then folding up the sides. Let $x$ be the side of the square cutouts. Then tell (a) what formula in terms of $x$ gives the volume of this box, (b) whether this a quadratic formula, and (c) what's the domain of $x$-values relevant to this problem.

Solution

According to the diagram below, the dimensions of the box are $x$ by $8-2x$ by $5-2x$

(a) The volume is $V = x (8 - 2x)(15-2x) = 120x - 46 x^2 + 4 x^3$.

(b) This is a cubic rather than a quadratic.

(c) The relevant values of $x$ are $0 < x < 4$. If $x$ is outside these limits then the box disappears, that is, it has a length negative or zero in some direction.

Note: A standard calculus problem is to find the value of $x$ that maximizes the volume of the box. Calculus is good for finding maximums or minimums of things. You could also set the problem up on a spreadsheet to hone in on the answer.