1. **State the problem:** We need to find the area of a composite figure made of a rectangle and two semicircles attached to the shorter sides of the rectangle. 2. **Identify given dimensions:** The rectangle has length $15$ units and height $10$ units. The height $10$ units is also the diameter of each semicircle. 3. **Formula for area of rectangle:** $$\text{Area}_{rectangle} = \text{length} \times \text{height}$$ 4. **Calculate area of rectangle:** $$\text{Area}_{rectangle} = 15 \times 10 = 150$$ 5. **Formula for area of a circle:** $$\text{Area}_{circle} = \pi r^2$$ 6. **Calculate radius of semicircle:** $$r = \frac{\text{diameter}}{2} = \frac{10}{2} = 5$$ 7. **Calculate area of one semicircle:** $$\text{Area}_{semicircle} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (5)^2 = \frac{1}{2} \pi 25 = \frac{25\pi}{2}$$ 8. **Calculate area of two semicircles (which equals one full circle):** $$2 \times \frac{25\pi}{2} = 25\pi$$ 9. **Calculate total area of the figure:** $$\text{Area}_{total} = \text{Area}_{rectangle} + \text{Area}_{two semicircles} = 150 + 25\pi$$ 10. **Approximate using $\pi \approx 3.1416$:** $$25 \times 3.1416 = 78.54$$ $$\text{Area}_{total} \approx 150 + 78.54 = 228.54$$ 11. **Round to nearest tenth:** $$228.5$$ **Final answer:** The area of the figure is approximately $228.5$ square units.