A storage company needs to design a new storage box that has twice the volume of its largest box. Its largest box is 5 feet long, 4 feet wide, and 3 feet high.

Question

A storage company needs to design a new storage box that has twice the volume of its largest box. Its largest box is 5 feet long, 4 feet wide, and 3 feet high. The new box must be formed by increasing each dimension by the same amount. Find the increase in each dimension.

Answer 1

The volumes of two similar objects are in ratio as the cube of the scale factor.

For example, if a box has all of its dimensions doubled (x2) then the volume is 2³, or eight times as large.

For the given box to have a doubled volume the scale factor must be ³√2, or 1.25992105.

Length: 5(1.25992105) or 6.26660

Width: 4(1.25992105) or 5.03968

Height: 3(1.25992105) or 3.77976

 

 

Answer 2

Let x be the increase in each dimension.

 

The new volume will be (x+5)(x+4)(x+3), which must be two times the volume of the formerly largest box, or 2·5·4·3 = 120 ft³.

 

Solving (x+5)(x+4)(x+3) = 120,

 

(x² + 9x + 20)(x + 3) = 120

 

x³ +12x² + 47x + 60 = 120

 

x³ + 12x² + 47x - 60 = 0

 

A solution to this is x = 1. The increase in each dimension is 1 foot.