Question

一个半径为 $4$ 英寸的圆的四分之三扇形连同内部，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：圆锥展开图（立体几何）勾股定理

难度评级：1140

解答：

圆锥体积公式为 $r$ 。还要注意，圆锥底面周长等于原圆剩下的弧长，也就是完整圆周的 $r=3$ 。 $2\pi r=\frac34(2\pi\cdot4)=6\pi,$ $h=\sqrt{4^2-3^2}=\sqrt7.$

因此底面周长为：这说明圆锥底面半径为：粘合后，扇形半径成为圆锥母线长，所以圆锥高为：因此圆锥体积为：所以答案是 C。 $\frac13\pi r^2h=\frac13\pi(3)^2\sqrt7=3\pi\sqrt7.$

The sector's arc becomes the base circumference, so if the base radius is $r,$ then $2\pi r=\frac34(2\pi\cdot4)=6\pi,$ giving $r=3.$ The sector radius becomes the cone's slant height, so the cone's height is $h=\sqrt{4^2-3^2}=\sqrt7.$

Hence the volume is $\frac13\pi r^2h=\frac13\pi(3)^2\sqrt7=3\pi\sqrt7.$ Thus, C is the correct answer.

2020 AMC 10B 第 10 题

答案：C

难度评级：1140

解答：

其他年份的第 10 题