1. **Problem statement:** We have a solid made of a conical frustum with a hemispherical top. Given: - Top radius of frustum and hemisphere $r=3.5$ cm - Bottom radius of frustum $R=4.2$ cm - Slant height of frustum $l=8$ cm - Frustum height $h$ cm (unknown) We need to find: (a) Surface area of the solid (b) Radius of base for a similar solid with surface area 81.51 cm² (c) (i) Height $h$ of the frustum (ii) Volume of the solid --- 2. **Formulas and rules:** - Surface area of conical frustum (lateral only): $$A_{frustum} = \pi (R + r) l$$ - Surface area of hemisphere (curved surface only): $$A_{hemisphere} = 2 \pi r^2$$ - Total surface area of solid (excluding base of frustum since hemisphere covers top): $$A_{total} = A_{frustum} + A_{hemisphere} + \text{area of base circle} = \pi (R + r) l + 2 \pi r^2 + \pi R^2$$ - Height of frustum from Pythagoras: $$h = \sqrt{l^2 - (R - r)^2}$$ - Volume of frustum: $$V_{frustum} = \frac{1}{3} \pi h (R^2 + Rr + r^2)$$ - Volume of hemisphere: $$V_{hemisphere} = \frac{2}{3} \pi r^3$$ - Total volume: $$V = V_{frustum} + V_{hemisphere}$$ --- 3. **Calculate surface area (a):** Given $\pi = \frac{22}{7}$ Calculate lateral surface area of frustum: $$A_{frustum} = \pi (R + r) l = \frac{22}{7} (4.2 + 3.5) \times 8 = \frac{22}{7} \times 7.7 \times 8$$ Calculate: $$\frac{22}{7} \times 7.7 = 22 \times 1.1 = 24.2$$ So, $$A_{frustum} = 24.2 \times 8 = 193.6 \text{ cm}^2$$ Calculate hemisphere surface area: $$A_{hemisphere} = 2 \pi r^2 = 2 \times \frac{22}{7} \times (3.5)^2 = 2 \times \frac{22}{7} \times 12.25$$ Calculate: $$\frac{22}{7} \times 12.25 = 22 \times 1.75 = 38.5$$ So, $$A_{hemisphere} = 2 \times 38.5 = 77 \text{ cm}^2$$ Calculate base area: $$A_{base} = \pi R^2 = \frac{22}{7} \times (4.2)^2 = \frac{22}{7} \times 17.64$$ Calculate: $$\frac{22}{7} \times 17.64 = 22 \times 2.52 = 55.44$$ Total surface area: $$A_{total} = 193.6 + 77 + 55.44 = 326.04 \text{ cm}^2$$ --- 4. **Find height $h$ of frustum (c)(i):** Use Pythagoras: $$h = \sqrt{l^2 - (R - r)^2} = \sqrt{8^2 - (4.2 - 3.5)^2} = \sqrt{64 - 0.7^2} = \sqrt{64 - 0.49} = \sqrt{63.51}$$ Calculate: $$h \approx 7.97 \text{ cm}$$ --- 5. **Find volume of solid (c)(ii):** Calculate volume of frustum: $$V_{frustum} = \frac{1}{3} \pi h (R^2 + Rr + r^2) = \frac{1}{3} \times \frac{22}{7} \times 7.97 \times (4.2^2 + 4.2 \times 3.5 + 3.5^2)$$ Calculate inside parentheses: $$4.2^2 = 17.64, \quad 4.2 \times 3.5 = 14.7, \quad 3.5^2 = 12.25$$ Sum: $$17.64 + 14.7 + 12.25 = 44.59$$ Calculate volume: $$V_{frustum} = \frac{1}{3} \times \frac{22}{7} \times 7.97 \times 44.59$$ Calculate stepwise: $$\frac{22}{7} \times 7.97 = 22 \times 1.1386 = 25.05$$ Then: $$V_{frustum} = \frac{1}{3} \times 25.05 \times 44.59 = \frac{1}{3} \times 1117.5 = 372.5 \text{ cm}^3$$ Calculate volume of hemisphere: $$V_{hemisphere} = \frac{2}{3} \pi r^3 = \frac{2}{3} \times \frac{22}{7} \times (3.5)^3 = \frac{2}{3} \times \frac{22}{7} \times 42.875$$ Calculate: $$\frac{22}{7} \times 42.875 = 22 \times 6.125 = 134.75$$ So, $$V_{hemisphere} = \frac{2}{3} \times 134.75 = 89.83 \text{ cm}^3$$ Total volume: $$V = 372.5 + 89.83 = 462.33 \text{ cm}^3$$ --- 6. **Find radius of base for similar solid with surface area 81.51 cm² (b):** For similar solids, all linear dimensions scale by factor $k$. Surface area scales by $k^2$. Given original surface area $A_1 = 326.04$ cm² and new surface area $A_2 = 81.51$ cm², $$k^2 = \frac{A_2}{A_1} = \frac{81.51}{326.04} = 0.25$$ So, $$k = \sqrt{0.25} = 0.5$$ Radius of base scales by $k$: $$R_{new} = k R = 0.5 \times 4.2 = 2.1 \approx 2 \text{ cm (nearest whole number)}$$