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An open box

CalculusMedium

From a rectangular sheet of card $18$ cm by $48$ cm, equal squares of side $x$ cm are cut from the $4$ corners and the sides are folded up to make an open box. What is the largest possible volume of the box, in cm³? Give the answer as an integer.

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Hint

Write the volume as a cubic in $x$ and keep only the critical point that leaves a real box.

Solution

The box has height $x$ , width $18 - 2 x$ and length $48 - 2 x$ , so $V (x) = x (18 - 2 x) (48 - 2 x) = 4 x^{3} - 132 x^{2} + 864 x$ for $0 < x < 9$ .
Then $V^{'} (x) = 12 x^{2} - 264 x + 864 = 12 (x - 4) (x - 18)$ .
Only $x = 4$ lies in the allowed range, and $V$ is $0$ at the ends of the range, so it gives the maximum.
The box is $4$ by $10$ by $40$ , so the largest volume is $4 \times 10 \times 40 = 1600$ .

Answer: $1600$

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