2020 AMC 12B Problem 9

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 volume Pythagorean Theorem

Difficulty rating: 1470

Small hint:

The arc length of the sector becomes the circumference of the cone’s base

Big hint:

The sector radius

4

becomes the slant height; find the base radius, then the cone’s height

Solution:

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.