Find the dimensions of the resulting box that has the largest volume.

Question

An open box is to be made out of a 12-inch by 18-inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. Find the dimensions of the resulting box that has the largest volume.

Answer 1

If x is the height of the box, the base will be 12-2x x 18-2x

Then, the volume is x(18-2x)(12-2x) = 4x^3 - 60x^2 + 216x

We maximize by taking the derivative and setting it equal to 0.

12x^2 - 120x + 216 = 0

12(x^2 - 10x + 18) = 0

12(x - 5 + √7)(x - 5 - √7) = 0

0 <= x <= 6 for the problem

Thus, x = 5 - √7

The dimension of the box is 18 - 2x by 12 - 2x by x = 8 + 2√7 by 2 + 2√7 by 5 - √7