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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Let the side length of the squares $x$ feet.
First, we want to determine the dimensions of the rectangular box in terms of $x$. We cut out four squares from the corners, so the length of the base of the box will be $16-2x$ feet, and the width will be $12-2x$ feet. The height of the box is $x$ feet.
The volume $V$ of the box is given by $V=(16-2x)(12-2x)(x)=4x^3-56x^2+192x$.
To maximize $V$, we need to find the critical points. Taking the derivative, we have $V' = 12x^2 - 112x + 192$. Setting this equal to 0 and solving for $x$, we get $4x^2 - 37x + 64 = 0$. Factorizing the quadratic, we have $(4x-1)(x-64) = 0$.
Hence, either $x = \frac{1}{4}$ or $x = 64$. However, $x$ must be between 0 and the length of the side it is cut from, so $0 < x < \frac{1}{4}$. Therefore, the only critical point within the feasible range is $x = \boxed{\frac{1}{4}}$ feet.
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