1. **Problem statement:** Find the area of the shaded region in a large circle with center $Q$ and area $9\pi$ square inches, where three equal semicircles are drawn on the diameter of the large circle. 2. **Given data:** - Area of the large circle $Q$ is $9\pi$. - Three equal semicircles are drawn on the diameter of the large circle. 3. **Find the radius of the large circle:** The area of a circle is given by the formula: $$\text{Area} = \pi r^2$$ Given: $$9\pi = \pi r^2$$ Divide both sides by $\pi$: $$\cancel{\pi}9 = \cancel{\pi} r^2 \implies 9 = r^2$$ Taking the square root: $$r = 3$$ 4. **Diameter of the large circle:** $$\text{Diameter} = 2r = 2 \times 3 = 6$$ 5. **Radius of each small semicircle:** Since three equal semicircles are drawn on the diameter, each semicircle has diameter: $$\frac{6}{3} = 2$$ So radius of each semicircle is: $$r_{small} = \frac{2}{2} = 1$$ 6. **Area of one small semicircle:** Area of a semicircle is half the area of a full circle: $$A_{semicircle} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (1)^2 = \frac{\pi}{2}$$ 7. **Total area of three small semicircles:** $$3 \times \frac{\pi}{2} = \frac{3\pi}{2}$$ 8. **Area of the shaded region:** The shaded region is the area of the large circle minus the area covered by the three small semicircles: $$9\pi - \frac{3\pi}{2} = \frac{18\pi}{2} - \frac{3\pi}{2} = \frac{15\pi}{2}$$ **Final answer:** $$\boxed{\frac{15\pi}{2}}$$ square inches