1. **Problem Statement:** We have a solid composed of a conical frustum with a hemispherical top. Given: - Top radius $r = 3.5$ cm - Bottom radius $R = 4.2$ cm - Slant height of frustum $l = 8$ cm - Frustum height $h$ cm (unknown) - Use $\pi = \frac{22}{7}$ We need to find: (a) Surface area of the solid (b) Radius of base of a similar solid with surface area $81.51$ cm$^2$ (c) (i) Height $h$ of the frustum (ii) Volume of the solid 2. **Formulas and Important Rules:** - Surface area of conical frustum (lateral) = $\pi (R + r) l$ - Surface area of hemisphere = $2 \pi r^2$ - Total surface area = lateral surface area of frustum + hemisphere surface area + base area of frustum - Base area of frustum = $\pi R^2$ - Height of frustum $h$ relates to slant height $l$ by Pythagoras: $$h = \sqrt{l^2 - (R - r)^2}$$ - Volume of frustum = $\frac{1}{3} \pi h (R^2 + Rr + r^2)$ - Volume of hemisphere = $\frac{2}{3} \pi r^3$ 3. **(a) Find surface area of the solid:** - 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, $A_{frustum} = 24.2 \times 8 = 193.6$ cm$^2$ - Calculate surface area of hemisphere: $$A_{hemisphere} = 2 \pi r^2 = 2 \times \frac{22}{7} \times (3.5)^2 = 2 \times \frac{22}{7} \times 12.25$$ Calculate $\frac{22}{7} \times 12.25 = 22 \times 1.75 = 38.5$ So, $A_{hemisphere} = 2 \times 38.5 = 77$ cm$^2$ - Calculate base area of frustum: $$A_{base} = \pi R^2 = \frac{22}{7} \times (4.2)^2 = \frac{22}{7} \times 17.64$$ Calculate $\frac{22}{7} \times 17.64 = 22 \times 2.52 = 55.44$ cm$^2$ - Total surface area: $$A_{total} = A_{frustum} + A_{hemisphere} + A_{base} = 193.6 + 77 + 55.44 = 326.04 \text{ cm}^2$$ 4. **(b) Find radius of base of similar solid with surface area 81.51 cm$^2$:** - Similar solids scale surface area by square of scale factor $k^2$. - Original surface area $= 326.04$, new surface area $= 81.51$ - Find scale factor: $$k^2 = \frac{81.51}{326.04} = 0.25 \Rightarrow k = 0.5$$ - Radius of base scales by $k$: $$R_{new} = k \times R = 0.5 \times 4.2 = 2.1 \approx 2 \text{ cm (nearest whole number)}$$ 5. **(c) (i) Find height $h$ of frustum:** - Use Pythagoras: $$h = \sqrt{l^2 - (R - r)^2} = \sqrt{8^2 - (4.2 - 3.5)^2} = \sqrt{64 - 0.7^2} = \sqrt{64 - 0.49} = \sqrt{63.51}$$ $$h \approx 7.97 \text{ cm}$$ 6. **(c) (ii) Find volume of the solid:** - 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 $\frac{22}{7} \times 7.97 = 22 \times 1.1386 = 25.05$ So, $$V_{frustum} = \frac{1}{3} \times 25.05 \times 44.59 = \frac{1}{3} \times 1116.5 = 372.17 \text{ cm}^3$$ - 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: $$V_{total} = V_{frustum} + V_{hemisphere} = 372.17 + 89.83 = 462 \text{ cm}^3$$ **Final answers:** - (a) Surface area = $326.04$ cm$^2$ - (b) Radius of base of similar solid = $2$ cm - (c) (i) Height $h = 7.97$ cm - (c) (ii) Volume = $462$ cm$^3$