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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 $=\sqrt{r^{2}+h^{2}}$
$=\sqrt{(5)^{2}+(12)^{2}}$
$=\sqrt{25+144}$
$=\sqrt{169}$
$=13 \mathrm{~m}$