Question #26

Question

The side of one cube is 2 cm more than the side of another cube. The volumes of the cubes differ by 152cm^3. Find the lengths of the sides of each cubes. 

Answer 1

Hi Lauren, the volume of a cube is: V = s³. Let x be the length of the side of one cube, then x + 2 is the length of the side of the other cube.

 

Given the difference in the volume of the two cubes: 152 cubic centimeters.

 

Volume of cube one:

 

V = x³

 

Volume of cube two:

 

V = (x + 2)³ = x³ + 6x² + 12x + 8

 

The difference in volumes is given by:

 

x³ + 6x² + 12x + 8 - x³ = 152

 

Collect terms gives:

 

6x² + 12x + 8 = 152

 

Subtract 8 from both sides:

 

6x² + 12x = 144

 

Divide both sides by 6

 

x² + 2x = 144 / 6, or 72 / 3

 

Complete the square on the left giving:

 

x² + 2x + 1 = 72 / 3 + 1

 

(x + 1)² = 75 / 3, (note 1 = 3 / 3, add the 3 / 3 to the 72 / 3 giving 75 / 3)

 

Take the square root of both sides:

 

x + 1 = ±√(75 / 3) = ±5

 

Subtract 1 from both sides:

 

x = ±5 - 1

 

Solution:

 

x = 4, or x = -6

 

We can discard the solution x = -6 since the length of a side of the cube cannot be negative.

 

Check:

 

x³ = 4³ = 64 cubic centimeters

 

(x + 2)³ = (4 + 2)³ = 216 cubic centimeters

 

Difference:

 

216 - 64 = 152 cubic centimeters

 

Answer to question:

 

Length of cube one is 4 centimeters.

 

Length of cube two is 6 centimeters.

 

Questions?