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.

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

Solution:

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:

$\frac{4}{3} \pi\left[(R)^{3}-(r)^{3}\right]$

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

Therefore,
Volume of the hollow sphere =  Volume of the cone formed

$\frac{4}{3} \pi\left[(4)^{3}-(2)^{3}\right]=\frac{1}{3} \pi \times(8)^{2} \times \mathrm{h}$

$\Rightarrow 4(64-8)=64 \times \mathrm{h}$

$\Rightarrow 224=64 \times \mathrm{h}$

$\Rightarrow \mathrm{h}=\frac{224}{64}$

$\Rightarrow \mathrm{h}=3.5 \mathrm{~cm}$

 

 

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