Question

A metallic spherical shell, of internal and external diameters 4 cm and 8 cm, is melted and recast as 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

(b) 14 cm
Let the internal and external radii of the spherical shell be r and R, respectively.
Then, $R\Rightarrow \frac{8}{2}\text{cm}=4\text{cm},r\Rightarrow \frac{4}{2}=2\text{cm}$
Volume of the spherical shell$=\frac{4}{3}\mathrm{\pi }\left[R^3-r^3\right]$
$=\frac{4}{3}\mathrm{\pi }\left[{\left(4\right)}^3-{\left(2\right)}^3\right]$
Volume of the cone$=\frac{1}{3}\mathrm{\pi }{r}^{2}h$
$=\frac{1}{3}\mathrm{\pi }\times {\left(4\right)}^2\times h$
Therefore,

$$\begin{gathered} \frac{4}{3}\mathrm{\pi }\left[{\left(4\right)}^3-{\left(2\right)}^3\right]=\frac{1}{3}\mathrm{\pi }\times {\left(4\right)}^2\times \mathrm{h} \\ \Rightarrow 4\times 56=16\times \mathrm{h} \\ \Rightarrow h=\frac{56}{4} \\ \Rightarrow h=14\text{cm} \end{gathered}$$