Question

A box (with no top) is to be constructed from a piece of cardboard of sides a and b by cutting out squares of length h from the corners and folding up the sides. find the value of h that maximizes the volume of the box if a = 8 and b = 9.

Answer 1

The volume of the box is h(8-2h)(9-2h). Differentiating this gives dV/dh = 12h^2-68h+72. The derivative is zero when h = (17+sqrt(73))/6 or (17-sqrt(73))/6. The second derivative is positive for the first value, so this gives a minimum volume. The second derivative is negative for the second value, so this gives the maximum volume. So the value of h that maximizes the volume of the box is (17-sqrt(73))/6, which is approximately 1.41.