Maximum Volume of a Cone
A circular disc of radius R is used to make a Cone by removing a sector with angle ø and then joining the edges where the sector was removed. What is the maximum volume that such a cone can attain? That is, what angle ø for a disc of radius R would you remove to make a cone of maximum volume?
Because the circumference will miss a sector the circumference C = 2pR - qR. Now to find the radius r of the cone when the ends of the sectors are joined, we have
2pR - qR = 2pr
R - qR/2p = r
R(1 - q/2p) = r
In order to find the volume of the cone, we will need the height of the cone.
Let us construct the height of the cone, h, which is perpendicular to r.
h = sqrt(R2 - r2)
h = sqrt[R2 - R2(1 - q/p + q2/4p2))]
h = R sqrt(q/p - q2/4p2)
Now, we write the volume V as function of q, while R is fixed.
V = pr2h/3
V(q) = p( R(1 - q/2p))2(R sqrt(q/p - q2/4p2))/3
V’(q) = 1/3(pR3)(( 2p - q)/2p2)[ 1/sqrt(q/p - q2/4p2)]
Let V’(q) = 0 and we have
q = 2p, at which point the volume is zero. To find the optimum volume let us compare our answers in a spreadsheet.