Question

Two solid cylinders are $12cm$ and $18cm$ respectively in height. Their diameters are $12cm$ and $16cm$ respectively. Both the cylinders are melted and the material is molded into a solid right circular cone of height $33cm$. Find its diameter.

Answer 1

For cylinder $1$

Height $h_1=12cm$

Diameter $=12cm$

Radius $r_1=6cm$

For cylinder $2$

Height $h_2=18cm$

Diameter $=16cm$

Radius $r_2=8cm$

For cone

Height $h=33cm$

Let the radius be $r$

Total volume remains same

Volume of cylinder $1+$Volume of cylinder $2=$Volume of cone

$\Rightarrow \pi r_1^2{h}_1+\pi r_2^2{h}_2=\frac{1}{3}\left(\pi r^{2}h\right)$

$\Rightarrow \left(6{)}^2\right(12)+(8{)}^2\left(18\right)=\frac{1}{3}\left(r^2\right)\left(33\right)$

$\Rightarrow 36\times 12+64\times 18=11r^2$

$\Rightarrow 1584=11r^2$

$\Rightarrow r^2=\frac{1584}{11}$

$\Rightarrow r^2=144$

$\Rightarrow r=\sqrt{144}$

$\Rightarrow r=12$

$\therefore Diameter=2\times 12cm=24cm.$

Hence, the diameter of the cone is $24cm$