Question

A cone of height 24cm has a curved surface area 550 $c{m^2}$. Find its volume.

Answer 1

Hint:
We will start by finding the radius of the cone by substituting the value of $\pi $ and slant height, which is $l = \sqrt {{r^2} + {h^2}} $ in the formula of curved surface of area, $\pi rl$. Then, substitute the values of perpendicular height, radius and $\pi $ in the formula of volume.

Complete step by step solution:
We are given that the surface area of the cone is 550$c{m^2}$
We also know that curved surface of a cone is calculated using the formula $\pi rl$, where $r$ is the radius of the cone, $l$ is the slant height of the cone represented by $l = \sqrt {{r^2} + {h^2}} $
We are also given that the height of the cone is 24cm.
Then, on substituting the values of curved surface area and height in the formula of the cone.
$
  A = \pi r\left( {\sqrt {{r^2} + {h^2}} } \right) \\
  550 = \pi r\left( {\sqrt {{r^2} + {{\left( {24} \right)}^2}} } \right) \\
$
Now, find the value of $r$, substitute the value of $\pi = \dfrac{{22}}{7}$ in the above equation and solving it further.
$
  550 = \dfrac{{22}}{7}r\left( {\sqrt {{r^2} + 576} } \right) \\
   \Rightarrow r\left( {\sqrt {{r^2} + 576} } \right) = \dfrac{{550}}{{22}} \times 7 \\
   \Rightarrow r\left( {\sqrt {{r^2} + 576} } \right) = 25 \times 7 \\
$
On squaring both sides, we get,
$
  {r^2}\left( {{r^2} + 576} \right) = {25^2} \times {7^2} \\
   \Rightarrow {r^4} + 576{r^2} - \left( {625 \times 49} \right) = 0 \\
   \Rightarrow {r^4} + 625{r^2} - 49{r^2} - \left( {625 \times 49} \right) = 0 \\
   \Rightarrow \left( {{r^2} + 625} \right)\left( {{r^2} - 49} \right) = 0 \\
$
Equate each factor to 0
Then,
${r^2} = - 625$, which is not possible as square of every number is positive.
And
$
  {r^2} = 49 \\
  r = 7 \\
$
Thus, the radius of the cone is 7cm
But, we have to find the volume of the cone.
As it is known that the volume of the cone is given by $V = \dfrac{1}{3}\pi {r^2}h$, where $r$ is the radius and $h$ is the height of the cone.
On substituting the values of radius and height in the formula of volume, we get,
$
  V = \dfrac{1}{3}\left( {\dfrac{{22}}{7}} \right){\left( 7 \right)^2}\left( {24} \right) \\
   \Rightarrow V = \left( {22} \right)\left( 7 \right)\left( 8 \right) \\
   \Rightarrow V = 1232c{m^3} \\
$

Hence, the volume of the cone is \[1232c{m^3}\]

Note:
The volume of the cone is the amount of space enclosed by the cone. Students must remember the formulas of curved surface area and volume of the cone. The area is measured in square units and volume is measured in cubic units.