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

(Take π = 3.14)

1. 14 cm

2. 16 cm

3. 24 cm

4. 26 cm

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

Let r = 5x and h = 12x

Volume of cone = $\dfrac{1}{3}$ πr²h

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

⇒ r = 5x = 5 × 2 = 10 cm

⇒ h = 12x = 12 × 2 = 24 cm

Curved surface area = πrl

l² = r² + h²

⇒ l² = 10² + 24²

⇒ l² = 100 + 576

⇒ l² = 676

⇒ l = $\sqrt{676}$ = 26 cm

Hence, option 4 is the correct option.