Question

If the volume of a cone is $1212{\text{c}}{{\text{m}}^3}$ and its height is $24{\text{cm}}$. Find the surface area of the cone. $\left( {\pi = \dfrac{{22}}{7}} \right)$

Answer 1

Hint: In this question firstly we have to mention what is given to us, it will give a clear picture of what we have to find then, apply the required formula. Here we have to apply a volume of cones to get the value of $r$ . And then we will get our desired answer.

Complete step-by-step answer:
Here we have given that,

Volume of cone $ = 1212{\text{c}}{{\text{m}}^3}$
And height of cone $ = 24{\text{cm}}$
Also we know that,

Volume of a cone $ = \pi {r^2}\dfrac{h}{3}$
Also we have given, volume of cone $ = 1212$
$ \Rightarrow 1212 = \pi {r^2}\dfrac{h}{3}$ -----(1)

Now on solving the above equation we will get the value of $r$
Also we have given that the value of $\pi = \dfrac{{22}}{7}$ and $h = 24$
Now putting the values of $\pi $ and $h$ in (1) we get,
$1212 = \dfrac{{22}}{7} \times {r^2} \times 24 \times \dfrac{1}{3}$
Or $\dfrac{{1212 \times 7}}{{22 \times 8}} = {r^2}$
Or $\dfrac{{8484}}{{176}} = {r^2}$
Or ${r^2} = 48.20$
Or $r = 6.9429{\text{cm}}$
Thus, the value of $r$ is $6.9429{\text{cm}}$
Now we have to find the curved surface area of the cone.

And we know that the curved surface area of the cone $ = \pi rl$ ------(2)
And the value of slant height,$l = \sqrt {{r^2} + {h^2}} $ ------(3)
Now putting the value of $r$ and $h$ in (3) we get,
$l = \sqrt {{{\left( {6.9429} \right)}^2} + {{\left( {24} \right)}^2}} $
Or $l = \sqrt {48.20 + 576} $
Or $l = 24.984$

Now putting the value of (3) in (2) we get,
Surface area of cone $ = \pi rl$
$ = \dfrac{{22}}{7} \times 6.9429 \times 24.984$
Or Surface area of cone $ = 545.1{\text{c}}{{\text{m}}^2}$
Thus, the surface area of the cone is $545.1{\text{c}}{{\text{m}}^2}$.

Note: Whenever we face such types of questions the key concept is that we should write what is given to us, like we did. Then we apply the formula of volume of cone, from there we find the value of radius of cone then simply apply the surface area of the cone and thus we get our answer.