2009 AMC 10A Problem 11

Question

One dimension of a cube is increased by

1,

another is decreased by

1,

and the third is left unchanged. The volume of the new rectangular solid is

5

less than that of the cube. What was the volume of the cube?

8

27

64

125

216

Let the cube have side

x

and write the new volume as

x(x+1)(x-1)

Note that

x(x+1)(x-1) = x^3 - x

Let the cube have side length

x.

The new solid has volume

x(x+1)(x-1) = x^3 - x.

Setting this equal to

x^3 - 5

gives

x^3 - x = x^3 - 5,

so

x = 5.

The cube’s volume is

5^3 = 125.

Thus, the correct answer is D.

Problem 11 in other years