First of all before reading this...draw the geometry of the condition given....i am assuming that u have drawn a sphere a cone ...with
Slant height = x radius of cone = r and heigh of cone = h
Radiius of sphere = R after this read my solution
The height of the cone h = R + (something)
something = distance of centre from the base of the cone....you need to find this something okk
Let it be z
then by applying a few strength on mind you will get ..
z = root( R^2 - r^2) [ because there is a small right angle triangle formed by side R , r and z with R as largest side )
Now surface area S = π.r.x= f(x)
again x , h and r form a rt.angled triangle..
x^2 = h^2 + r^2 => x^2 = (R+z)^2 + r^2 = R^2 + z^2 + 2.z.R + r^2
x^2 = R^2 + R^2 -r^2 + 2.root(R^2-r^2).R + r^2
x^2 = 2.R^2 + 2.root(R^2-r^2).R
now put x=R
R^2 = 2.R^2 + 2.root(R^2 -r^2).R
-R^2 = 2.root(R^2 -r^2).R
-R = 2.root( R^2 -r^2)
R^2 = 4.(R^2 - r^2)
4r^2 = 3R^2
or r = R.root(3)/2 --------(1)
f(x) = pi.x.r
So f(R) = pi.R.( R.root(3)/2 ) = root(3).pi.R^2 / 2 ans ....
If u r unable to understand my steps...then please ask furthure...