Answer :

We know that total surface area of the cone = S = πr(l + r) …(1)

and Volume of the cone(V) =



Then by (1), we get,




P = V2


Now, differentiating P with respect to r, we get,



Now, if, , then


S = 4πr2


Now again differentiating with respect to r, we get


Therefore, P is maximum when S = 4πr2


And V is maximum when S = 4πr2


πr(l + r) = 4πr2


l = 3r


SinƟ =



Therefore, semi-vertical angle of right circular cone of given surface area and maximum volume is .


Hence Proved.


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