Question

What is the maximum volume in cubic inches of an open box to be made from a 12-inch by 16-inch piece of cardboard by cutting out squares of equal sides from the four corners and bending up the sides? Your work must include a statement of the function and its derivative. Give one decimal place in your final answer. (10 points)

Answer 1

194.07 cubic inches.

Explanation:

The cardboard is 12x16 before removing a square from each end. This square is x inches wide. Thus the 16 inside is shortened by x inches on both sides, or it is now 16-2x inches. The 12 inside is also reduced by 2x. The x value is also the height of the box when you fold the sides up. Thus the volume V = wlh = (16-2x)*(12-2x)*(x) = 4x^3 - 56x^2 + 192x.

To find the maximum, take the derivative, and find its roots

V = 4x^3 - 56x^2 + 192x

dV/dx = 12x^2 - 112x + 192

The roots are (14+2(13)^.5)/3 ~= 7.07 and (14-2(13)^.5)/3 ~= 2.26

The roots would be the possible values of x, the square we cut. Since 7.07 x 2 = 14.14 inches, this exceeds the 12 inch side, thus x = 2.26 inches. Thus you cut 2.26 inches from each corner to obtain the maximum volume.

Cube is 11.48 x 7.48 x 2.26 with a volume of 194.07 cubic inches.