2008 AMC 10B 第 19 题

Question

一个圆柱形水箱半径为

4

英尺，高为

9

英尺，横放在地上。水箱中水深为

2

英尺。水的体积是多少立方英尺？

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?

24\pi-36\sqrt{2}

24\pi-24\sqrt{3}

36\pi-36\sqrt{3}

36\pi-24\sqrt{2}

48\pi-36\sqrt{3}

答案：E

知识点：圆柱扇形体积

难度评级：1680

小提示：

水的横截面是由一条低于圆心

2

英尺的弦切出的圆弓形。

The water cross-section is a circular segment cut by a chord

2

feet below the center

大提示：

该面积等于一个

120^\circ

扇形减去一个三角形；再乘以长度

9

。

Its area is a

120^\circ

sector minus a triangle; multiply by the length

9

解答：

被水浸没的横截面是圆弓形。弦在圆心下方

4-2=2

英尺，且

\cos\theta=\tfrac{2}{4}=\tfrac12

，所以半角为

60^\circ

，圆心角为

120^\circ

。

扇形面积为

\tfrac{120}{360}\pi(4)^2=\tfrac{16\pi}{3}

，两条半径形成的三角形面积为

\tfrac12(4)^2\sin 120^\circ=4\sqrt3

，所以圆弓形面积为

\tfrac{16\pi}{3}-4\sqrt3

。

乘以长度

9

，体积为

9\left(\tfrac{16\pi}{3}-4\sqrt3\right)=48\pi-36\sqrt3

。

所以正确答案是 E。

The submerged cross-section is a circular segment. The chord is

4-2=2

feet below the center, and

\cos\theta=\tfrac{2}{4}=\tfrac12,

so the half-angle is

60^\circ

and the central angle is

120^\circ.

The sector area is

\tfrac{120}{360}\pi(4)^2=\tfrac{16\pi}{3},

and the triangle formed by the two radii has area

\tfrac12(4)^2\sin 120^\circ=4\sqrt3.

The segment area is

\tfrac{16\pi}{3}-4\sqrt3.

Multiplying by the length

9

gives

9\left(\tfrac{16\pi}{3}-4\sqrt3\right)=48\pi-36\sqrt3.

Thus, the correct answer is E.

2008 AMC 10B 第 19 题

答案：E

难度评级：1680

小提示：

大提示：

解答：

其他年份的第 19 题