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How to maximize the volume of an open box cut from a square

The model: height times base area

If the original square has side $L$ and each corner cutout has side $x$ , the folded box has height $x$ and square base side $L - 2 x$ . Its volume is $V (x) = x (L - 2 x)^{2}$ for $0 < x < L /2$ . Check interior critical points and the domain endpoints before naming a maximum.

Problem

A 15-inch by 15-inch sheet of cardboard is made into an open box by cutting a square of side $x$ from each corner and folding up the sides. Find $V (x)$ , the cut size that maximizes the volume, and the largest volume.

Answer

V (x) = x (15 - 2 x)^{2}

; cut

2.5

-inch squares; maximum volume

250

cubic inches.

The 7.5-inch stationary point makes a flat box and is outside the usable domain.

Step-by-step solution

1. Express the volume

The height is $x$ . Each base side loses $x$ at both ends, so it measures $15 - 2 x$ . Thus

V (x) = x (15 - 2 x)^{2}, 0 < x < 7.5.

2. Find critical points

Differentiate the factored expression:

V^{'} (x) = (15 - 2 x) (15 - 6 x) .

The derivative vanishes at $x = 2.5$ and $x = 7.5$ . The latter is a boundary where the base disappears. The derivative changes from positive to negative at $2.5$ , so this cut gives the maximum.

3. Calculate the volume

The box has a $10$ -inch square base and height $2.5$ inches:

V (2.5) = 2.5 (15 - 5)^{2} = 2.5 \cdot 100 = 250 in^{3} .

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