A metallic cone of base radius 2.1 cm and height 8.4 cm is melted and moulded into a sphere.
Question:

A metallic cone of base radius 2.1 cm and height 8.4 cm is melted and moulded into a sphere. The radius of the sphere is
(a) 2.1 cm
(b) 1.05 cm
(c) 1.5 cm
(d) 2 cm

Solution:

(a) 2.1 cm
Radius of cone = 2.1 cm
Height of cone = 8.4 cm

Volume of cone $=\frac{1}{3} \pi \mathrm{r}^{2} \mathrm{~h}=\frac{1}{3} \pi(2.1)^{2} \times 8.4$

Volume of the sphere $=\frac{4}{3} \pi r^{3}$

Therefore,

Volume of cone = Volume of sphere

$\Rightarrow \frac{1}{3} \pi \times(2.1)^{2} \times 8.4=\frac{4}{3} \pi r^{3}$

$\Rightarrow 4.41 \times 8.4=4 r^{3}$

$\Rightarrow 37.044=4 r^{3}$

$\Rightarrow r^{3}=\frac{37.044}{4}$

$\Rightarrow r^{3}=(2.1)^{3}$

$\Rightarrow r=2.1 \mathrm{~cm}$

Hence, the radius of the sphere is 2.1 cm.