Question

一个半径为 $4$ 英寸的圆的四分之三扇形及其内部，可以沿图中所示的两条半径粘合卷成一个直圆锥的侧面。这个圆锥的体积是多少立方英寸？

A three-quarter sector of a circle of radius $4$ inches together with its interior can be rolled up to form the lateral surface of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches?

$3\pi \sqrt{5}$

$4\pi \sqrt{3}$

$3\pi \sqrt{7}$

$6\pi \sqrt{3}$

$6\pi \sqrt{7}$

答案：C

知识点：圆锥体积勾股定理

难度评级：1470

解答：

扇形弧长为 $\tfrac34 \cdot 2\pi \cdot 4 = 6\pi$ 它等于底面周长。因此 $2\pi r = 6\pi$ 所以 $r = 3$ 。

母线长是 $4$ ，因此由勾股定理可得高为 $h = \sqrt{4^2 - 3^2} = \sqrt{7}$ 。圆锥体积为 $\frac13 \pi r^2 h = \frac13 \pi \cdot 9 \cdot \sqrt{7} = 3\pi\sqrt{7}.$

所以正确答案是 C。

The sector's arc length is $\tfrac34 \cdot 2\pi \cdot 4 = 6\pi,$ which becomes the base circumference: $2\pi r = 6\pi,$ so $r = 3.$

The slant height is the sector radius $4,$ so the height is $h = \sqrt{4^2 - 3^2} = \sqrt{7}.$ The volume is $\frac13 \pi r^2 h = \frac13 \pi \cdot 9 \cdot \sqrt{7} = 3\pi\sqrt{7}.$

Thus, the correct answer is C.

2020 AMC 12B 第 9 题

答案：C

难度评级：1470

解答：

其他年份的第 9 题