1. Let's find the surface area and volume of the first shape: a right cylinder with two hemispheres on its ends. 2. The cylinder has radius $r=6.0$ cm and height $h=8.5$ cm. Each hemisphere has radius $r=6.0$ cm. 3. Volume formulas: - Cylinder volume: $$V_{cyl} = \pi r^2 h$$ - Hemisphere volume: $$V_{hem} = \frac{2}{3} \pi r^3$$ 4. Surface area formulas: - Cylinder lateral surface area (side only): $$A_{cyl} = 2 \pi r h$$ - Hemisphere surface area (outer curved surface only): $$A_{hem} = 2 \pi r^2$$ 5. Since the hemispheres cover the cylinder's circular ends, the total surface area is the cylinder's side plus the two hemispheres' curved surfaces: $$A_{total} = A_{cyl} + 2 \times A_{hem} = 2 \pi r h + 2 \times 2 \pi r^2 = 2 \pi r h + 4 \pi r^2$$ 6. Calculate volumes: - Cylinder volume: $$V_{cyl} = \pi \times 6.0^2 \times 8.5 = \pi \times 36 \times 8.5 = 306 \pi$$ - Two hemispheres volume (which make one full sphere): $$2 \times V_{hem} = 2 \times \frac{2}{3} \pi 6.0^3 = \frac{4}{3} \pi 216 = 288 \pi$$ - Total volume: $$V_{total} = 306 \pi + 288 \pi = 594 \pi$$ 7. Calculate surface area: $$A_{total} = 2 \pi \times 6.0 \times 8.5 + 4 \pi \times 6.0^2 = 102 \pi + 144 \pi = 246 \pi$$ 8. Use $\pi \approx 3.1416$: - Volume: $$594 \pi \approx 594 \times 3.1416 = 1866.1 \text{ cm}^3$$ - Surface area: $$246 \pi \approx 246 \times 3.1416 = 772.8 \text{ cm}^2$$ --- 9. Now for the second shape: a right square prism and a right square pyramid attached. 10. Prism dimensions: base $1.0 \times 1.0$ m, height $2.0$ m. 11. Pyramid base length $1.0$ m, slant height $1.5$ m. 12. Volume formulas: - Prism volume: $$V_{prism} = \text{base area} \times \text{height} = 1.0 \times 1.0 \times 2.0 = 2.0 \text{ m}^3$$ - Pyramid volume: $$V_{pyr} = \frac{1}{3} \times \text{base area} \times \text{height}$$ 13. Find pyramid height using Pythagoras theorem: $$\text{height} = \sqrt{1.5^2 - (\frac{1.0}{2})^2} = \sqrt{2.25 - 0.25} = \sqrt{2.0} \approx 1.414 \text{ m}$$ 14. Pyramid volume: $$V_{pyr} = \frac{1}{3} \times 1.0 \times 1.0 \times 1.414 = 0.471 \text{ m}^3$$ 15. Surface area formulas: - Prism surface area: $$A_{prism} = 2 \times \text{base area} + \text{perimeter} \times \text{height} = 2 \times 1.0 + 4 \times 1.0 \times 2.0 = 2 + 8 = 10 \text{ m}^2$$ - Pyramid lateral area (4 triangles): each triangle area $$= \frac{1}{2} \times \text{base} \times \text{slant height} = 0.5 \times 1.0 \times 1.5 = 0.75$$ - Total lateral area $$= 4 \times 0.75 = 3.0$$ - Pyramid total surface area $$= \text{base area} + \text{lateral area} = 1.0 + 3.0 = 4.0 \text{ m}^2$$ 16. Since the pyramid is attached to the prism on one face, subtract one base area from total surface area: $$A_{total} = A_{prism} + A_{pyr} - \text{shared face} = 10 + 4 - 1 = 13 \text{ m}^2$$ 17. Total volume: $$V_{total} = V_{prism} + V_{pyr} = 2.0 + 0.471 = 2.5 \text{ m}^3$$ --- **Final answers:** - a) Surface area $\approx 772.8$ cm$^2$, Volume $\approx 1866.1$ cm$^3$ - b) Surface area $= 13.0$ m$^2$, Volume $\approx 2.5$ m$^3$