2020 AMC 10B Problem 10

Problem 10 of 25EasierGeometry

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?

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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.

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