A right circular cone is 3.6 cm high and the radius of its base is 1.6 cm. It is melted and recast into a right circular cone with radius of its base as 1.2 cm. Find its height.

Radius of cone, r = 1.6 cm

Height of the cone, h = 3.6 cm

Volume of circular cone = $\dfrac{1}{3}$ πr²h

$= \dfrac{1}{3} \times \dfrac{22}{7} \times (1.6)^2 \times 3.6 \\[1em] = \dfrac{1}{3} \times \dfrac{22}{7} \times 2.56 \times 3.6 \\[1em] = \dfrac{202.752}{21}$

Volume of cone of radius (R) = 1.2 cm and height (H)

Volume of circular cone = $\dfrac{1}{3}$ πR²H

$= \dfrac{1}{3} \times \dfrac{22}{7} \times (1.2)^2 \times \text{H} \\[1em] = \dfrac{1}{3} \times \dfrac{22}{7} \times 1.44 \times \text{H} \\[1em] = \dfrac{31.68}{21} \text{H}$

Since, cone is melted and recasted into right circular cone of radius 1.2 cm, the volume remains the same.

$\therefore \dfrac{202.752}{21} = \dfrac{31.68}{21} \text{H} \\[1em] \Rightarrow \text{H} = \dfrac{202.752 \times 21}{31.68 \times 21} \\[1em] \Rightarrow \text{H} = 6.4 \text{ cm.}$

Hence, height of the cone is 6.4 cm.