AMC10 2008 B

AMC10 2008 B · Q19

AMC10 2008 B · Q19. It mainly tests 3D geometry (volume).

A cylindrical tank with radius 4 feet and height 9 feet is lying on its side. The tank is filled with water to a depth of 2 feet. What is the volume of the water, in cubic feet?

一个半径 4 英尺、高 9 英尺的圆柱形水箱侧卧着。水箱中水深 2 英尺。水的体积是多少立方英尺？

(A) $24\pi -36\sqrt{2}$ $24\pi -36\sqrt{2}$

(B) $24\pi -24\sqrt{3}$ $24\pi -24\sqrt{3}$

(D) $36\pi -24\sqrt{2}$ $36\pi -24\sqrt{2}$

(E) $48\pi -36\sqrt{3}$ $48\pi -36\sqrt{3}$

Answer

Correct choice: (E)

正确答案：（E）

Solution

Answer (E): The portion of each end of the tank that is under water is a circular sector with two right triangles removed as shown. The hypotenuse of each triangle is 4, and the vertical leg is 2, so each is a $30-60-90^\circ$ triangle. Therefore the sector has a central angle of $120^\circ$, and the area of the sector is $$\frac{120}{360}\cdot \pi(4)^2=\frac{16}{3}\pi.$$ The area of each triangle is $\frac12(2)(2\sqrt3)$, so the portion of each end that is underwater has area $\frac{16}{3}\pi-4\sqrt3$. The length of the cylinder is 9, so the volume of the water is $$9\left(\frac{16}{3}\pi-4\sqrt3\right)=48\pi-36\sqrt3.$$

答案（E）：水箱两端在水下的部分如图所示，是一个圆扇形去掉两个直角三角形后的区域。每个三角形的斜边为 4，竖直直角边为 2，因此每个都是$30-60-90^\circ$三角形。所以该扇形的圆心角为$120^\circ$，扇形面积为 $$\frac{120}{360}\cdot \pi(4)^2=\frac{16}{3}\pi.$$ 每个三角形的面积是$\frac12(2)(2\sqrt3)$，因此每个端面在水下的面积为$\frac{16}{3}\pi-4\sqrt3$。圆柱的长度为 9，所以水的体积为 $$9\left(\frac{16}{3}\pi-4\sqrt3\right)=48\pi-36\sqrt3.$$

Topics

GE.013 3D geometry (volume)