1985 AMC 12 Problem 25

Question

The volume of a certain rectangular solid is

8\text{ cm}^3,

its total surface area is

32\text{ cm}^2,

and its three dimensions are in geometric progression. The sum of the lengths in cm of all the edges of this solid is

28

32

36

40

44

Write the three dimensions as

\frac{t}{r},t,tr

Use the volume to find

t,

then use the surface area to find

r+\frac{1}{r}

Write the dimensions as

\frac{2}{r},2,2r,

since their product is

8.

Half the surface area is the sum of the three pairwise products, so

\frac4r+4+4r=16,

giving

r+\frac{1}{r}=3.

The sum of the dimensions is

2(r+1+\frac{1}{r})=8.

Since each dimension occurs on four edges, the sum of all edge lengths is

4\cdot8=32.

Thus the correct answer is B.

Problem 25 in other years