In the diagram, two circles, each with centre $D$, have radii of 1 and 2. The total area of the shaded regions is $\frac{5}{12}$ of the area of the larger circle. What is a possible measure of $\angle A D C$ ?
(A) $108^{\circ}$
(B) $120^{\circ}$
(C) $90^{\circ}$
(D) $150^{\circ}$
(E) $135^{\circ}$

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-101.jpg?height=322&width=314&top_left_y=262&top_left_x=1293)
The total area of the larger circle is $\pi\left(2^{2}\right)=4 \pi$, so the total area of the shaded regions must be $\frac{5}{12}(4 \pi)=\frac{5}{3} \pi$.

Suppose that $\angle A D C=x^{\circ}$.

The area of the unshaded portion of the inner circle is thus $\frac{x}{360}$ of the total area of the inner circle, or $\frac{x}{360}\left(\pi\left(1^{2}\right)\right)=\frac{x}{360} \pi\left(\right.$ since $\angle A D C$ is $\frac{x}{360}$ of the largest possible central angle $\left(360^{\circ}\right)$ ). The area of the shaded portion of the inner circle is thus $\pi-\frac{x}{360} \pi=\frac{360-x}{360} \pi$.

The total area of the outer ring is the difference of the areas of the outer and inner circles, or $\pi\left(2^{2}\right)-\pi\left(1^{2}\right)=3 \pi$.

The shaded area in the outer ring will be $\frac{x}{360}$ of this total area, since $\angle A D C$ is $\frac{x}{360}$ of the largest possible central angle $\left(360^{\circ}\right)$.

So the shaded area in the outer ring is $\frac{x}{360}(3 \pi)=\frac{3 x}{360} \pi$.

So the total shaded area (which must equal $\frac{5}{3} \pi$ ) is, in term of $x, \frac{3 x}{360} \pi+\frac{360-x}{360} \pi=\frac{360+2 x}{360} \pi$. Therefore, $\frac{360+2 x}{360}=\frac{5}{3}=\frac{600}{360}$, so $360+2 x=600$ or $x=120$.

Thus, $\angle A D C=120^{\circ}$.

ANSWER: (B)