Q23 of 168 Page 231

Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is of the volume of the sphere.

Let r and h be the radius and height of the cone respectively inscribed in a sphere of radius R.

Let V be the volume of cone.


Then, V=


And height of cone h = R +



Now,







Now, if,


After solving this we get,


So, when, , then < 0


Then, by second derivative test, the volume of the cone is the maximum when


So, when, , h = R +


Therefore, V =



Therefore, the volume of the largest cone that can be inscribed in the sphere is the volume of the sphere.


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