A cone-shaped drinking cup is made from a cirular piece of paper of radius R by cutting out a sector and joining the edge CA and CB. Find the maximum capacity of such a cup.
The answer is attached.
Let the height of the cone be h and of the sector it is formed from be R.
Then the radius of the base of the cone is
r = sqrt(R^2-h^2)
and so the volume of the cone is:
V = (1/3) h pi r^2 = (pi/3) h (R^2-h^2)
This volume is a maximum when dV/dh = 0, but:
dV/dh = (pi/3) (R^2 - 3 h^2),
which is zero when h = R/3, which is clearly a maximum, so the maximum
volume V = (8 pi/27) R^3.
RonL