Find the dimensions of the box of maximum volume.

Question

A rectangular box is to be made from a piece of cardboard 20 inches long and 8 inches wide by cutting out identical squares from the four corners and turning up the sides

Find the dimensions of the box of maximum volume.

Length:  inches

width:  inches

height:  inches.

What is this volume?  cubic inches

Answer 1

Let x(0 < x < 4). is side of squares. Then volume V = (20 - 2x)(8 - 2x)x = 160x - 56x² + 4x³;

V' = 160 - 112x + 12x² = 0; 3x² - 28x + 40 = 0; x = (14 - √196 - 120)/3 = (14 - 2√19)/3 ≈ 1.761

V'' = 24x - 112; V''(1.761) = - 69.736 < 0. So, we have max volume at x = 1.761

Length: 20 - 2x = 16.478 inches

width: 8 - 2x = 4.478 inches

height: x = 1.761 inches