calculus on dimensions of a box

Question

An open box is to be made out of a 8-inch by 16-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

Taking x from both sides of the box and turning this up makes the dimensions of the box to be

V = (8-2x)(16-2x)x for the Volume calculation.

Finding the maximum valued, we'll take the derivative and set it equal to zero to solve for the maximum value of x.

V = (8-2x)(16-2x)x = 128x - 48x² + 4x³

V' = 128 - 96x + 12x² = 0

V' = 32 - 24x + 3x² cancelling 4 from the equation

Solving for x either graphing or with the quadratic formula yields

x = 1.69 or 6.30

6.30 is eliminated because it is too large to have 12.6 subtracted from 8 for one side of the box.

Therefore the box has dimensions of 4.62 x 12.62 x 1.69 inches