Optimization: A cone-shaped drinking cup

Question

A cone-shaped drinking cup is made from a circular piece of paper of radius R by cutting out a sector and joining the edges CA and CB. Find the maximum capacity of such a cup (Your answer may depend on R).

Answer 1

Volume of the cone is

V=πr²h/3

So we have to find the values of r and h in terms of R. r is the radius of the circle of the top of the cone and h is the height of the cone.

R is the slant height of the cone which is the hypotenuse of the right triangle.

R² = h²+r² or r²=R²-h²

So V = (1/3)π[R²-h²]h = (π/3)[R²h-h³]

To find the maximum volume of the cone, need to take the derivative of the Volume and set it equal to zero and solve for h.

dV/dh=(π/3)[R²-3h²]

R²-3h²=0 --> h²=R²/3 -> h = R/√3

Now we can solve for r:

r²=R²-R²/3 = (2/3)R²

Substituting for r and h into the volume formula:

V = (π/3)r²h = (π/3)(2R²/3)(R/√3)

V(R)=2πR³/(9√3)