A covered box is to be made from a rectangular sheet of card

A covered box is to be made from a rectangular sheet of cardboard measuring 35 inches by 56 inches. This done by cutting out the shaded regions of the figure and then folding on the dotted lines. What are the dimensions x, y, and z that maximize the volume? x = inches y = inches z = inches

Solution

In the figure

2x + 2y = 56 => x + y = 28 => y = 28 - x

z + 2x = 35 => z = 35 - 2x

Volume ( v ) = x * y * z = x * ( 28 - x ) * ( 35 - 2x ) = x ( 980 + 2 * x^2 - 91 * x ) = 2x^3 - 91 x^2 + 980 x

For maximum volume dv/dx = 0

=> 6 X^2 -182 X + 980 = 0 => X = 1/12 { 182 +- (33124 - 23520)^1/2 }

=> X = 70/3 & 7

Now find second derivative of v

d2v/ dx2 = 12 X - 182 = 280 -182 = 98 at X= 70/3

and = 84 - 182 = -98 at X = 7

at X = 7 d2v/dx2 < 0

therefore maximum value of v is for X = 7 Inches

Y = 28 - X = 28 - 7 = 21 Inches

Z = 35 - 2X = 35 - 14 = 21 Inches