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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.

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Solution

## 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}×\frac{22}{7}×{\left(4x\right)}^{2}×3\mathrm{x}$$⇒2156=\frac{1}{3}×\frac{22}{7}×16{x}^{2}×3\mathrm{x}$$⇒2156=\frac{22}{7}×16{x}^{3}$$⇒15092=352{x}^{3}$$⇒{x}^{3}=42.875$$⇒x=\sqrt[3]{42.875}$$⇒x=3.5cm$Therefore,The radius of the right circular cone $=4x=4×3.5=14cm$Height of the right circular cone $=3x=3×3.5=10.5cm$Hence, the radius and the height of the cone are $14cm$and $10.5cm$ respectively .

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