Question

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?

$3\pi\sqrt{5}$

$4\pi\sqrt{3}$

$3\pi\sqrt{7}$

$6\pi\sqrt{3}$

$6\pi\sqrt{7}$

Answer: C

Concepts:cone net (3D geometry)Pythagorean Theorem

Difficulty rating: 1140

Solution:

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 Problem 10

Answer: C

Difficulty rating: 1140

Solution:

Problem 10 in Other Years