2020 AMC 10B Problem 10

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

Small hint:

The sector arc becomes the circumference of the cone base

Big hint:

The sector radius becomes the slant height of the cone

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.