Question

A solid hemisphere of wax of radius $12\;cm$ is melted and made into a cone of base radius $6\;cm$. Calculate the height of the cone.

Answer 1

Hint: Calculate the volume of the solid hemisphere and also use the formula for the volume of the cone that is formed and equate both the volumes to find the height of the cone.

Complete step-by-step answer:
A solid hemisphere of wax of radius $12\;cm$ is melted and made into a cone of base radius $6\;cm$.
The formula for the volume of the hemisphere is equal to $\dfrac{2}{3}\pi {r^3}$.
As per given the radius of the hemisphere is equal to $12\;cm$. Now calculate the volume of the hemisphere.
$
\Rightarrow V = \dfrac{2}{3}\pi {r^3} \\
   = \dfrac{2}{3} \times \dfrac{{22}}{7} \times {\left( {12\;cm} \right)^3} \\
   = 3619.11\;c{m^3} \;
 $
The volume of the hemisphere is equal to $3619.11\;c{m^3}$.
Assume that the height of the cone is equal to $h$ and the radius of the cone is equal to $6\;cm$. So, the volume of the cone is equal to
$\pi {r^2}\dfrac{h}{3} = \dfrac{{22}}{7} \times {\left( {6\;cm} \right)^2} \times \dfrac{h}{3} = \dfrac{{264}}{7}h\;c{m^3}$.
The hemisphere transforms into a cone. So, the volume of both will be equal to each other.
$
\Rightarrow \dfrac{{264}}{7}h\;c{m^3} = 3619.11\;c{m^3} \\
  h = 3619.11 \times \dfrac{7}{{264}} \\
  h = 96.04\;cm \;
 $
So, the height of the cone is equal to $96.04\;cm$.
So, the correct answer is “$96.04\;cm$”.

Note: The solid hemisphere is transformed into a cone of certain height by melting it so the volume of both the figures will be equal to each other and the height will be obtained from that equation.