Question

Find the maximum volume of a cone that can be out of a solid hemisphere of radius \[r\].

Answer 1

Hint: In this question, first draw the diagram it will give us a clear picture of what we have to find out. Then insert the cone in such a way that the volume is maximum which will lead us to give the final answer by measuring the radius and height of the cone.

Complete step by step solution:
The radius of the hemisphere is \[r\] units.
The maximum volume of the cone that can be inserted in the given solid hemisphere has a radius and height of the cone is equal to the radius of the given hemisphere as shown in the below figure.

So, the radius of the cone is equal to \[r\] units and the height of the cone is equal to \[r\] units.
We know that the volume of the cone with radius \[r\] units and height \[h\] units is given by \[V = \dfrac{1}{3}\pi {r^2}h{\text{ cu}}{\text{.units}}\].
Therefore, volume of the required cone \[ = \dfrac{1}{3}\pi {r^2}\left( r \right){\text{ cu}}{\text{.units }} = \dfrac{1}{3}\pi {r^3}{\text{ cu}}{\text{.units}}\]
Thus, the volume of the required cone is \[\dfrac{1}{3}\pi {r^3}{\text{ cu}}{\text{.units}}\].

Note: The volume of the cone with radius \[r\] units and height \[h\] units is given by \[V = \dfrac{1}{3}\pi {r^2}h{\text{ cu}}{\text{.units}}\]. Remember that the maximum volume of the cone that can be inscribed in a hemisphere is equal to half of the volume of the hemisphere.