The radius and the height of a right circular cone are in the ratio 5 : 12 and its volume is 2512 cubic cm. Find the radius and slant height of the cone. (Take π = 3.14)

Given,

Radius : Height = 5 : 12

Let radius (r) = 5x and height (h) = 12x.

By formula,

Volume of cone = $\dfrac{1}{3}πr^2h$

$\Rightarrow 2512 = \dfrac{1}{3} \times 3.14 \times (5x)^2 \times (12x) \\[1em] \Rightarrow 2512 = \dfrac{1}{3} \times 3.14 \times 300x^3 \\[1em] \Rightarrow x^3 = \dfrac{2512 \times 3}{3.14 \times 300} \\[1em] \Rightarrow x^3 = \dfrac{7536}{942} \\[1em] \Rightarrow x^3 = 8 \\[1em] \Rightarrow x^3 = 2^3 \\[1em] \Rightarrow x = 2 \text{ cm}.$

Radius (r) = 5x = 5(2) = 10 cm and Height (h) = 12x = 12(2) = 24 cm.

By formula,

⇒ l² = h² + r²

⇒ l² = (24)² + (10)²

⇒ l² = 576 + 100

⇒ l² = 676

⇒ l = $\sqrt{676}$ = 26 cm.

Hence, radius = 10 cm and slant height = 26 cm.