1. **Problem Statement:** Find the value of $x$ given that the area of sector $OAPB$ is $\frac{5}{18}$ of the area of the circle with center $O$. 2. **Formula for sector area:** The area of a sector with central angle $x$ (in degrees) in a circle of radius $r$ is given by: $$\text{Area of sector} = \frac{x}{360} \times \pi r^2$$ 3. **Given:** $$\text{Area of sector } OAPB = \frac{5}{18} \times \text{Area of circle} = \frac{5}{18} \times \pi r^2$$ 4. **Set up equation:** $$\frac{x}{360} \times \pi r^2 = \frac{5}{18} \times \pi r^2$$ 5. **Simplify:** Cancel $\pi r^2$ on both sides: $$\frac{x}{360} = \frac{5}{18}$$ 6. **Solve for $x$:** $$x = 360 \times \frac{5}{18} = 360 \times \frac{5}{18}$$ Calculate: $$x = 360 \times \frac{5}{18} = 20 \times 5 = 100$$ 7. **Answer:** The value of $x$ is $100^\circ$. This means the central angle $x$ subtending the sector $OAPB$ is $100$ degrees.