A rectangular box with an open top is to be constructed from a 12-foot by 16-foot sheet of metal by cutting squares of equal size from the corners and folding up the sides. Determine the size of the squares to maximize the volume of the box.

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Topic: Differential Calculus Posted 1 year ago

A rectangular box with an open top is to be constructed from a 12-foot by 16-foot sheet of metal by cutting squares of equal size from the corners and folding up the sides. Determine the size of the squares to maximize the volume of the box.

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Koala Tutor

Oct 8, 2023

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Let $x$ be the side length of each square to be cut out. We cut 4 squares, one from each corner of length $x$. We then fold up the sides to form a box without a top. The dimensions of the resulting box are $12-2x$, $16-2x$, and $x$. The volume of the box is $(12-2x)(16-2x)(x)$. Applying the techniques from Solution 2 of this example question, we note that the maximum volume of $(12-2x)(16-2x)(x)$ on the interval $0

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