1. **Problem Statement:** We have a solid made 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 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 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 $h = \sqrt{l^2 - (R - r)^2}$ (from Pythagoras) - Volume of frustum = $\frac{1}{3} \pi h (R^2 + Rr + r^2)$ - Volume of hemisphere = $\frac{2}{3} \pi r^3$ --- 3. **(a) Surface Area Calculation:** - 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 intermediate: $$ \frac{22}{7} \times 7.7 = \frac{22}{7} \times \frac{77}{10} = \frac{22 \times 77}{7 \times 10} = \frac{1694}{70} $$ So, $$ A_{frustum} = \frac{1694}{70} \times 8 = \frac{1694 \times 8}{70} = \frac{13552}{70} = 193.6 \text{ cm}^2 $$ - 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 intermediate: $$ \frac{22}{7} \times 12.25 = \frac{22}{7} \times \frac{49}{4} = \frac{22 \times 49}{7 \times 4} = \frac{1078}{28} = 38.5 $$ So, $$ A_{hemisphere} = 2 \times 38.5 = 77 \text{ cm}^2 $$ - Base area of frustum: $$ A_{base} = \pi R^2 = \frac{22}{7} \times (4.2)^2 = \frac{22}{7} \times 17.64 $$ Calculate intermediate: $$ \frac{22}{7} \times 17.64 = \frac{22}{7} \times \frac{1764}{100} = \frac{22 \times 1764}{7 \times 100} = \frac{38808}{700} = 55.44 $$ - Total surface area: $$ A_{total} = A_{frustum} + A_{hemisphere} + A_{base} = 193.6 + 77 + 55.44 = 326.04 \text{ cm}^2 $$ --- 4. **(b) Radius of base for similar solid with surface area 81.51 cm²:** - Surface area scales with square of linear dimensions. - Let scale factor be $k$, then: $$ k^2 = \frac{81.51}{326.04} = 0.25 $$ - So, $$ k = \sqrt{0.25} = 0.5 $$ - Radius of base of similar solid: $$ R_{new} = k \times R = 0.5 \times 4.2 = 2.1 \approx 2 \text{ cm (nearest whole number)} $$ --- 5. **(c) (i) Height $h$ of frustum:** - Use Pythagoras: $$ h = \sqrt{l^2 - (R - r)^2} = \sqrt{8^2 - (4.2 - 3.5)^2} = \sqrt{64 - 0.7^2} = \sqrt{64 - 0.49} = \sqrt{63.51} $$ Calculate: $$ h \approx 7.97 \text{ cm} $$ --- 6. **(c) (ii) 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 intermediate: $$ \frac{22}{7} \times 7.97 = \frac{22}{7} \times \frac{797}{100} = \frac{22 \times 797}{700} = \frac{17534}{700} = 25.05 $$ So, $$ V_{frustum} = \frac{1}{3} \times 25.05 \times 44.59 = \frac{1}{3} \times 1116.47 = 372.16 \text{ cm}^3 $$ - 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 intermediate: $$ \frac{22}{7} \times 42.875 = \frac{22}{7} \times \frac{342}{8} = \frac{22 \times 342}{7 \times 8} = \frac{7524}{56} = 134.36 $$ So, $$ V_{hemisphere} = \frac{2}{3} \times 134.36 = 89.57 \text{ cm}^3 $$ - Total volume: $$ V_{total} = V_{frustum} + V_{hemisphere} = 372.16 + 89.57 = 461.73 \text{ cm}^3 $$