AMC10 2017 A

AMC10 2017 A · Q11

AMC10 2017 A · Q11. 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?

由线段 AB 所有三维空间中距离不超过 3 个单位的所有点的区域，其体积为 \(216\pi\)。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。圆柱体的体积为 \(\pi \cdot 3^2 \cdot h = 9\pi h\)，两个半球的总积分为 \(\frac{4}{3}\pi \cdot 3^3 = 36\pi\)。因此 \(9\pi h + 36\pi = 216\pi\)，得 \(h = 20\)。

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