2024 AMC 8 Problem 18

Question

Three concentric circles centered at

O

have radii of

1,

2,

and

3.

Points

B

and

C

lie on the largest circle. The region between the two smaller circles is shaded, as is the portion of the region between the two larger circles bounded by central angle

BOC,

as shown in the figure below. Suppose the shaded and unshaded regions are equal in area. What is the measure of

\angle BOC

in degrees?

108

120

135

144

150

Answer: A

Concepts:annulus sector linear equation

Difficulty rating: 1540

Small hint:

Compute the area of the shaded annulus between radii

1

and

2

Big hint:

Set shaded area equal to unshaded area and solve for the sector angle

Solution:

Let

\theta

be the measure of

\angle BOC.

One component of the shaded region is the area of the circle with radius

2

minus the area of the circle with radius

1.

This part has area

4\pi-\pi=3\pi.

The remaining area is a sector of the biggest circle minus the area of the circle with radius

2

. This has area

\dfrac{\theta}{360}(9\pi-4\pi) = \dfrac{\theta}{360}(5\pi).

Hence, the total area of the shaded region is

3\pi + \dfrac{\theta}{360}(5\pi).

Next, we note that the unshaded region is composed of the smallest circle and the unshaded portion of the outer ring. This will have a total area of

\pi + \dfrac{360-\theta}{360}(5\pi).

Lastly, we equate the area of both regions and solve for

\theta:

\begin{gathered} 3\pi + \dfrac{\theta}{360}(5\pi) \\ = \pi + \dfrac{360-\theta}{360}(5\pi) \end{gathered}

2\pi = \dfrac{360-\theta-\theta}{360}(5\pi)

\dfrac{2}{5} = 1 - \dfrac{2\theta}{360}

2\theta = \dfrac{3}{5}(360)

\theta = 108.

Thus, A is the correct answer.