Question

A hollow sphere of internal and external diameters 4 cm and 8 cm respectively is melted into a cone of base diameter 8 cm. The height of the cone is

(a) 12 cm

(b) 14 cm

(c) 15 cm

(d) 18 cm

Answer 1

External radius

Internal radius

The volume of hollow sphere

$$\begin{gathered} V=\frac{4}{3}\mathrm{\pi }\left({\mathrm{R}}^3-{\mathrm{r}}^3\right) \\ =\frac{4}{3}\mathrm{\pi }\left(4^3-2^3\right) \end{gathered}$$

Let h be the height of cone.

Clearly,

The volume of recasted cone = volume of hollow sphere

$$\begin{gathered} \frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}=\frac{4}{3}\mathrm{\pi }\left(4^3-2^3\right) \\ \Rightarrow 4^2\mathrm{h}=4\left(4^3-2^3\right) \\ \Rightarrow \mathrm{h}=14\mathrm{cm} \end{gathered}$$

Hence, the height of cone = 14 cm

Hence, the correct answer is choice (b).