Solution:Let us first draw the diagram.
Let the height of the cone be and the radius of the cone be . From the diagram, we can see that forms a right-angled triangle. Since represents the radius of the sphere, therefore .
Also, represents the radius of the sphere, therefore .
As we can see that and , therefore
In right angles triangle .
Therefore, using Pythagoras’ Theorem,
As we can see ABAB represents the radius of the base of the cone, therefore .
Now the volume of the cone is given by ,
where is the radius of the base of the cone and is the height of the cone.
Let us take the derivative of with respect to . We will get,
Taking will give us two values of one of them gives the maximum volume of the cone and the other one gives the minimum volume of the cone.
gives and that is .
Taking derivative of , we get
When we put in , we get
,which is negative.
Therefore, represents the maximum volume of cones inscribed in the sphere.