1. **State the problem:** Solve for the variable $x$ in the equation $\sin^{-1}\left(\frac{11}{15}\right) = x$ and find the measure of angle $C$ (denoted as $m\angle C$). Given the right triangle relationship $11^2 + y^2 = 15^2$, find $y$ and then $m\angle C$. 2. **Formula and rules:** The inverse sine function $\sin^{-1}$ gives the angle whose sine is the given ratio. For a right triangle, $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$. Here, $\sin(x) = \frac{11}{15}$. 3. **Calculate $x$:** $$ x = \sin^{-1}\left(\frac{11}{15}\right) $$ Using a calculator, $$ x \approx 0.7854 \text{ radians} \approx 45^\circ $$ 4. **Find $y$ using Pythagoras:** $$ 11^2 + y^2 = 15^2 $$ $$ 121 + y^2 = 225 $$ $$ y^2 = 225 - 121 = 104 $$ $$ y = \sqrt{104} = 2\sqrt{26} \approx 10.198 $$ 5. **Interpretation:** The side $y$ is approximately 10.198 units. 6. **Find $m\angle C$:** Since $x = m\angle C = \sin^{-1}\left(\frac{11}{15}\right)$, the measure of angle $C$ is approximately $45^\circ$. **Final answer:** $$ m\angle C \approx 45^\circ $$