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

$28$

$32$

$36$

$40$

$44$

答案：B

知识点：等比数列长方体表面积体积

难度评级：1920

小提示：

将三个尺寸写成 $\frac{t}{r},t,tr$

Write the three dimensions as $\frac{t}{r},t,tr$

大提示：

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 $r+\frac{1}{r}$

解答：

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.

1985 AMC 12 第 25 题

答案：B

难度评级：1920

小提示：

大提示：

解答：

其他年份的第 25 题