Question

The volume of a cone is 396 ${\text{m}}{{\text{m}}^3}$. If the height of the cone is 10.5 mm, calculate the radius of its base. $\left( {\pi = \dfrac{{22}}{7}} \right)$

Answer 1

Hint: Write the formula of the volume of the cone. We need to substitute the value of the volume, height and use $\left( {\pi = \dfrac{{22}}{7}} \right)$ in the formula of the volume of the cone, $V = \dfrac{1}{3}\pi {r^2}h$ , where $V$ is the volume of the cone, $r$ is the radius of the base and $h$ is the height of the base.
Solve the equation to find the value of radius. Mention the unit of the radius after solving it.

Complete step by step answer:
As it is known that the volume of the cone is equal to one-third of the product of the area of the circular base and perpendicular height.
Therefore, the volume of the cone is given by $V = \dfrac{1}{3}\pi {r^2}h$, where $V$ is the volume of the cone, $r$ is the radius of the base and $h$ is the height of the base.
Here, the height mentioned in the question is not the slant height.
Hence, we are given that $V = 396$ and $h = 10.5$.
We have to calculate the radius of the base by substituting $V = 396$, $h = 10.5$ and $\pi = \dfrac{{22}}{7}$ in the formula $V = \dfrac{1}{3}\pi {r^2}h$.
$396 = \dfrac{1}{3}\left( {\dfrac{{22}}{7}} \right){r^2}\left( {10.5} \right)$
Solve for the value of $r$.
${r^2} = \dfrac{{3 \times 396 \times 7}}{{22 \times 10.5}}$
Taking square-root both sides to solve the equation,
$
  r = \sqrt {\dfrac{{3 \times 396 \times 7}}{{22 \times 10.5}}} \\
  r = \sqrt {36} \\
  r = 6{\text{ mm}} \\
 $
Hence, the radius of the cone is 6mm.

Note: Use $\pi = \dfrac{{22}}{7}$ in this question to avoid difficult calculations. Also, use the correct formula of the volume of the cone to get the correct answer. While substituting the values, $h = 10.5$ is the perpendicular height and not the slant height of the cone. Mention the unit of the radius after solving it.