1985 AMC 12 第 25 题

Question

某长方体的体积为

8\text{ cm}^3

，总表面积为

32\text{ cm}^2

，且三个尺寸构成等比数列。该长方体所有棱长之和（单位：厘米）为

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

将三个尺寸写成

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

Write the three dimensions as

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

先用体积求

t

，再用表面积求

r+\frac{1}{r}

Use the volume to find

t,

then use the surface area to find

r+\frac{1}{r}

将三个尺寸写成

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

，因为它们的乘积为

8

。表面积的一半等于三个两两乘积之和，所以

\frac4r+4+4r=16\text{，}

得到

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

。三个尺寸之和为

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

。每个尺寸在四条棱上出现，所以所有棱长之和为

4\cdot8=32

。

所以正确答案是 B。

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.

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