Question

Jesse 把一个半径为十二的圆形纸片沿两条半径剪开，形成两个扇形，其中较小扇形的圆心角为一百二十度。他用每个扇形各做一个圆锥的侧面。较小圆锥与较大圆锥的体积之比是多少？

Jesse cuts a circular paper disk of radius 12 along two radii to form two sectors, the smaller having a central angle of 120 degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller cone to that of the larger?

$\dfrac{1}{8}$

$\dfrac{1}{4}$

$\dfrac{\sqrt{10}}{10}$

$\dfrac{\sqrt{5}}{6}$

$\dfrac{\sqrt{10}}{5}$

答案：C

知识点：圆锥扇形体积

难度评级：1930

解答：

每个扇形形成的圆锥母线长为 $12$ 。较小扇形的圆心角为 $120^\circ$ ，弧长为 $\dfrac13\cdot2\pi\cdot12=8\pi$ ，所以较小圆锥底面半径为 $4$ ，高为 $\sqrt{12^2-4^2}=8\sqrt2$ 。

较大扇形弧长为 $16\pi$ ，底面半径为 $8$ ，高为 $\sqrt{12^2-8^2}=4\sqrt5$ 。

体积比为 $\dfrac{\frac13\pi\cdot4^2\cdot8\sqrt2}{\frac13\pi\cdot8^2\cdot4\sqrt5}=\dfrac{\sqrt{10}}{10}.$

所以正确答案是 C。

Each sector forms a cone with slant height $12$ . The smaller sector has angle $120^\circ$ , so its arc length is $\dfrac13\cdot2\pi\cdot12=8\pi$ , giving base radius $4$ . Its cone height is $\sqrt{12^2-4^2}=8\sqrt2$ .

The larger sector has arc length $16\pi$ , giving base radius $8$ . Its cone height is $\sqrt{12^2-8^2}=4\sqrt5$ .

The volume ratio is $\dfrac{\frac13\pi\cdot4^2\cdot8\sqrt2}{\frac13\pi\cdot8^2\cdot4\sqrt5}=\dfrac{\sqrt{10}}{10}.$

Thus, C is the correct answer.

2012 AMC 10B 第 17 题

答案：C

难度评级：1930

解答：

其他年份的第 17 题