Question

A hollow metallic sphere with external diameter 8 cm and internal diameter 4 cm is melted and moulded into a cone of base radius 8 cm. The height of the cone is
(a) 12 cm
(b) 14 cm
(c) 15 cm
(d) 18 cm

Answer 1

DISCLAIMER : The answer to the question does not match the options given.

External diameter = 8 cm
Internal diameter = 4 cm
Let the external and internal radii of the hollow metallic sphere be R and r, respectively.
Then,
External radius = $\frac{8}{2}=4\mathrm{cm}$
Internal Radius = $\frac{4}{2}=2\mathrm{cm}$
Then, volume of the hollow sphere:
$\frac{4}{3}\mathrm{\pi }\left[{\left(R\right)}^3-{\left(r\right)}^3\right]$

$=\frac{4}{3}\mathrm{\pi }\left[{\left(4\right)}^3-{\left(2\right)}^3\right]$

Therefore,
Volume of the hollow sphere = Volume of the cone formed
$$\begin{gathered} \frac{4}{3}\mathrm{\pi }\left[{\left(4\right)}^3-{\left(2\right)}^3\right]=\frac{1}{3}\mathrm{\pi }\times {\left(8\right)}^2\times \mathrm{h} \\ \Rightarrow 4\left(64-8\right)=64\times \mathrm{h} \\ \Rightarrow 224=64\times \mathrm{h} \\ \Rightarrow \mathrm{h}=\frac{224}{64} \\ \Rightarrow \mathrm{h}=3.5\mathrm{cm} \end{gathered}$$