# Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the cone.

consider the given figure

Let H and R be the height and radius of the base of the cone ABC respectively.
suppose the radius and height of the cylinder inserted in the cone be r and h resp.
Now,
AS
so,

Let s be the curved surface area of the cylinder
so,

Now, differentiating w.r.t to r, we get :
Again differentiating w.r.t to r, we get :

for

Now
so, curved surface area s, of the cylinder is maximum at
hence, the radius of the right circular cylinder of greatest curved surface area which can be insulated in a given cone is half of that of the cone.

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