Question

Corners of a rectangular paper of size 50 ft x 20 ft are chopped off. shapes of removed portions are squares of equal size. if remaining part of the paper is folded to make an opened-top box, what should be the dimension of removed squares so that the volume of the box is largest?

Answer 1

Let the dimensions of the removed squares be x by x

The dimensions of the base of the box will be;

Length = 50 - 2x
Width = 20 - 2x

The volume;

V = Base area x height (height = x)

V = (50 - 2x)(20 - 2x)x

V = (1000 - 100x - 40x + 4x²)x
V = (1000 - 140x + 4x²)x
V = 1000x - 140x² + 4x³

At maximum volume, dV/dx = 0

dV/dx = 12x² - 280x + 1000 = 0

Using the quadratic formula,

a = 12, b = -280, c = 1000

x = [280+/- √(-280)² - 4(12)(1000)]/2(12)

x = 18.93 

or 

x = 4.40


So, the dimensions of the removed squares can either by 18.93 by 18.93 or 4.4 by 4.4