1. Problem: Find the volume of a composite solid consisting of a hemisphere on top of a cube, each side of the cube is 8 cm. 2. Formulae: - Volume of cube: $$V_{cube} = s^3$$ where $s$ is the side length. - Volume of hemisphere: $$V_{hemisphere} = \frac{2}{3}\pi r^3$$ where $r$ is the radius. 3. Calculate cube volume: $$V_{cube} = 8^3 = 512 \text{ cm}^3$$ 4. Calculate hemisphere volume: Radius $r = 8$ cm (same as cube side since hemisphere sits on top) $$V_{hemisphere} = \frac{2}{3}\pi (8)^3 = \frac{2}{3}\pi 512 = \frac{1024}{3}\pi \text{ cm}^3$$ 5. Total volume: $$V_{total} = V_{cube} + V_{hemisphere} = 512 + \frac{1024}{3}\pi \approx 512 + 1072.33 = 1584.33 \text{ cm}^3$$ --- 6. Problem: Find the volume of a cylinder with radius 8 ft and height 4 ft. 7. Formula: - Volume of cylinder: $$V = \pi r^2 h$$ 8. Calculate volume: Radius $r = \frac{16}{2} = 8$ ft (diameter 16 ft) Height $h = 4$ ft $$V = \pi (8)^2 (4) = \pi \times 64 \times 4 = 256\pi \approx 804.25 \text{ ft}^3$$ --- 9. Problem: Find the volume of a cone with base diameter 16 ft and slant height 6 ft. 10. Formula: - Volume of cone: $$V = \frac{1}{3}\pi r^2 h$$ 11. Find height $h$ using Pythagoras theorem: Radius $r = \frac{16}{2} = 8$ ft Slant height $l = 6$ ft $$h = \sqrt{l^2 - r^2} = \sqrt{6^2 - 8^2} = \sqrt{36 - 64} = \sqrt{-28}$$ Since $h$ cannot be imaginary, re-check dimensions: slant height must be greater than radius. Assuming slant height is 10 ft (likely typo), then: $$h = \sqrt{10^2 - 8^2} = \sqrt{100 - 64} = \sqrt{36} = 6 \text{ ft}$$ 12. Calculate volume: $$V = \frac{1}{3}\pi (8)^2 (6) = \frac{1}{3}\pi 64 \times 6 = 128\pi \approx 402.12 \text{ ft}^3$$