The ratio of the radius and the height of a solid metallic right circular cylinder is 7 : 27. This is melted and made into a cone of diameter 14 cm and slant height 25 cm. Find the height of the :

(a) cone

(b) cylinder

Given,

Diameter of cone = 14 cm

Radius of cone (r) = $\dfrac{\text{Diameter}}{2} = \dfrac{14}{2}$ = 7 cm

Slant height (l) = 25 cm

Ratio of the radius and the height of a solid metallic right circular cylinder is 7 : 27.

Radius of cylinder (R) = 7x

Height of cylinder (H) = 27x

(a) Let height of cone be h cm.

By formula,

⇒ l² = r² + h²

⇒ 25² = 7² + h²

⇒ 625 = 49 + h²

⇒ h² = 625 - 49

⇒ h² = 576

⇒ h = $\sqrt{576}$ = 24 cm.

Hence, height of cone = 24 cm.

(b) Given,

A solid metallic right circular cylinder is melted and made into a cone.

∴ Volume of cylinder = Volume of cone

⇒ πR²H = $\dfrac{1}{3}πr^2h$

⇒ R²H = $\dfrac{1}{3}r^2h$

⇒ (7x)² × 27x = $\dfrac{1}{3} \times 7^2 \times 24$

⇒ 1323x³ = $\dfrac{1176}{3}$

⇒ 1323x³ = 392

⇒ x³ = $\dfrac{392}{1323}$

⇒ x³ = $\dfrac{8}{27}$

⇒ x³ = $\Big(\dfrac{2}{3}\Big)^3$

⇒ x = $\dfrac{2}{3}$

Height of cylinder (H) = 27x = $27 \times \dfrac{2}{3}$ = 18 cm.

Hence, height of cylinder = 18 cm.