Question

The radius and height of right circular cone are in the ratio $4:3$ and its volume is $2156c{m}^3$. Find the curved surface area of the cone.

Answer 1

Find the radius and height of the cone.

Given, the radius and height of right circular cone are in the ratio $4:3$

The volume of the cone $V=2156c{m}^3$

Let the radius and height of cone be $4x$ and $3x$ respectively.

Volume of the cone $V=\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}$, where $r$ is radius and $h$ is height

$\therefore 2156=\frac{1}{3}\times \frac{22}{7}\times {\left(4x\right)}^2\times 3\mathrm{x}$

$\Rightarrow 2156=\frac{1}{3}\times \frac{22}{7}\times 16x^2\times 3\mathrm{x}$

$\Rightarrow 2156=\frac{22}{7}\times 16x^3$

$\Rightarrow 15092=352x^3$

$\Rightarrow x^3=42.875$

$\Rightarrow x=\sqrt[3]{42.875}$

$\Rightarrow x=3.5cm$

Therefore,

The radius of the right circular cone $=4x=4\times 3.5=14cm$

Height of the right circular cone $=3x=3\times 3.5=10.5cm$

Hence, the radius and the height of the cone are $14cm$and $10.5cm$ respectively .