The radius and the height of a right circular cone are in the ratio of 5 : 12 and its volume is 2512 cm³. Find:

(i) the radius and height of the cone

(ii) the curved surface area of the cone

(iii) the total surface area of the cone

(Take π = 3.14)

Given, radius(r) : height(h) = 5 : 12

(i) Let r = 5x and h = 12x

Volume of cone = $\dfrac{1}{3}$ πr²h

$\Rightarrow 2512 = \dfrac{1}{3} \times 3.14 \times (5\text{x})^2 \times 12\text{x} \\[1em] \Rightarrow 2512 = 3.14 \times 25\text{x}^2 \times 4\text{x} \\[1em] \Rightarrow \text{x}^3 = \dfrac{2512}{3.14 \times 25 \times 4} \\[1em] \Rightarrow \text{x}^3 = \dfrac{2512}{314} \\[1em] \Rightarrow \text{x}^3 = 8 \\[1em] \Rightarrow \text{x} = \sqrt[3]{8} \\[1em] \Rightarrow \text{x} = 2$

⇒ r = 5x = 5 × 2 = 10 cm

⇒ h = 12x = 12 × 2 = 24 cm

Hence, radius of the cone is 10 cm and height of the cone is 24 cm.

(ii) Curved surface area = πrl

l² = r² + h²

⇒ l² = 10² + 24²

⇒ l² = 100 + 576

⇒ l² = 676

⇒ l = $\sqrt{676}$ = 26 cm

$= 3.14 \times 10 \times 26 \\[1em] = 816.4 \text{ cm}^2$

Hence, curved surface area of the cone is 816.4 cm².

(iii) Total surface area = πr(l + r)

$= 3.14 \times 10(26 + 10) \\[1em] = 3.14 \times 10 \times 36 \\[1em] = 1130.4 \text{ cm}^2$

Hence, total surface area of the cone is 1130.4 cm².