The radius and the height of a right circular cone are in the ratio $5 : 12$. If its volume is $314$ cubic metre, find the slant height and the radius (Use $\pi = 3.14$).

 Given:

The radius and the height of a right circular cone are in the ratio $5 : 12$.

Its volume is $314$ cubic metres.

To do:

We have to find the slant height and the radius.

Solution:

Ratio of the radius and height of the cone $= 5 : 12$

Volume of the cone $= 314\ cm^3$

Let the radius $(r)$ be $5x$ and the height $(h)$ be $12x$
Therefore,

Volume of the cone $=\frac{1}{3} \pi r^{2} h$

$314=\frac{1}{3} \times 3.14 \times(5 x)^{2}(12 x)$

$314 \times 3=3.14(25 x^{2} \times 12 x)$

$\frac{314 \times 3}{3.14}=300 x^{3}$

$\frac{314 \times 3 \times 100}{314 \times 300}=x^{3}$

$x^{3}=(1)^{3}$

$\Rightarrow x=1$

This implies,

Radius $(r)=5 x$

$=5 \times 1$

$=5 \mathrm{~m}$

Height $(h)=12 x$

$=12 \times 1$

$=12 \mathrm{~m}$

Slant height $=\sqrt{r^{2}+h^{2}}$

$=\sqrt{(5)^{2}+(12)^{2}}$

$=\sqrt{25+144}$

$=\sqrt{169}$

$=13 \mathrm{~m}$

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Updated on: 10-Oct-2022

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