Question

三维空间中所有距离线段 $\overline{AB}$ 不超过 $3$ 个单位的点组成的区域体积为 $216\pi$ 。 $AB$ 的长度是多少？

The region consisting of all points in three-dimensional space within $3$ units of line segment $\overline{AB}$ has volume $216\pi.$ What is the length $AB?$

$6$

$12$

$18$

$20$

$24$

答案：D

知识点：体积圆柱球

难度评级：1540

解答：

回忆一下，距离一个点不超过固定距离 $r$ 的所有点组成一个球。

在线段的两个端点处，可以把区域看成各形成一个半球。

中间各点周围也会形成球，但它们会并入相邻的部分。

这说明中间部分形成半径为 $3$ 的圆柱。两个半球组成半径为 $3$ 的球，体积为因此圆柱体积为 $216 \pi - 36 \pi = 180 \pi$ 。底面积为 $9 \pi$ ，若 $h$ 是 $AB$ ，则体积为 $\dfrac{4}{3} \pi 3^3 = 27 \cdot \dfrac{4}{3} \pi = 36 \pi.$ $\begin{aligned} \pi 3^2 h &= 180\pi \\9h \pi &= 180 \pi \\ h &= 20. \end{aligned}$

所以正确答案是 D。

Recall that all the points at most a fixed distance $r$ away from a point form a sphere.

At the end points of this line segment, we can visualize two hemispheres being formed at each end.

All the points in the middle also have spheres forming around them, but they get merged into the ones right next to them.

This means that the middle section forms a cylinder with radius $3.$ The two hemispheres form a sphere with radius $3,$ and therefore a volume of $\dfrac{4}{3} \pi 3^3 = 27 \cdot \dfrac{4}{3} \pi = 36 \pi.$ This means that the cylinder has a volume of $216 \pi - 36 \pi = 180 \pi.$ We know the base area is $9 \pi,$ so if $h$ is $AB,$ then the volume is $\begin{aligned} \pi 3^2 h &= 180\pi \\9h \pi &= 180 \pi \\ h &= 20. \end{aligned}$

Thus, D is the correct answer.

2017 AMC 10A 第 11 题

答案：D

难度评级：1540

解答：

其他年份的第 11 题