, 2022

Since the shaded regions are equal, then when the unshaded sector
in the small circle is shaded, the area of the now fully shaded sector
of the larger circle must be equal to the area of the smaller
circle.

[[IMAGE0]]

The smaller circle has radius 1 and so has area $\pi \times 1^2 = \pi$. The larger circle has radius 3 and so has area $\pi \times 3^2 = 9\pi$. This means that the area of the shaded sector in the larger circle has area $\pi$, which means that it must be $\frac{1}{9}$ of the larger circle. This means that $\angle POQ$ must be $\frac{1}{9}$ of a complete circle, and so $\$\angle$ POQ = $\frac{1}{9} \times 360\degree = 40\degree$$. Thus,$x = 40$.

Since the shaded regions are equal in area, then when the
unshaded sector in the small circle is shaded, the area of the now fully
shaded sector of the larger circle must be equal to the area of the
smaller circle.

[[IMAGE0]]

The smaller circle has radius 1 and so it has area $\pi \times 1^2 = \pi$. The larger circle has radius 3 and so it has area $\pi \times 3^2 = 9\pi$. This means that the area of the shaded sector in the larger circle is $\pi$, which means that it must be $\frac{1}{9}$ of the larger circle. This means that $\angle POQ$ must be $\frac{1}{9}$ of a complete circular angle, and so $\$\angle$ POQ = $\frac{1}{9} \times 360^\circ = 40^\circ$$. Thus,$x = 40$.