AMC12 2017 A

AMC12 2017 A · Q8

AMC12 2017 A · Q8. It mainly tests 3D geometry (volume), 3D geometry (surface area).

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

由三维空间中所有距离线段 AB 不超过 3 个单位的点的区域体积为 $216\pi$。AB 的长度是多少？

(A) 6 6

(B) 12 12

(D) 20 20

(E) 24 24

Answer

Correct choice: (D)

正确答案：（D）

Solution

Let $h = AB$. The region consists of a solid circular cylinder of radius 3 and height $h$, together with two solid hemispheres of radius 3 centered at A and B. The volume of the cylinder is $\pi \cdot 3^2 \cdot h = 9\pi h$, and the two hemispheres have a combined volume of $\frac{4}{3}\pi \cdot 3^3 = 36\pi$. Therefore $9\pi h + 36\pi = 216\pi$, and $h = 20$.

设 $h = AB$。该区域由半径为 3、高为 $h$ 的实心圆柱体以及以 A 和 B 为圆心、半径为 3 的两个实心半球组成。圆柱体体积为 $\pi \cdot 3^2 \cdot h = 9\pi h$，两个半球总体积为 $\frac{4}{3}\pi \cdot 3^3 = 36\pi$。因此 $9\pi h + 36\pi = 216\pi$，得 $h = 20$。

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