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.

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- April 9th 2007, 01:50 AMcamherokidOptimization Problem5
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. - April 9th 2007, 03:52 AMCaptainBlack
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