POSSIBLE SOLUTION(S)

This appears to be a function of two variables but if one writes out all the information one finds there is but one variable, we used theta - the angle of the cut sector.

We offer a plot of a typical bounded region.

We define our variables:

We presume a sector of angle theta and a circle of radius r. We specify that the length of the wire has to be the sum of the two radii (the edges of the sector) and the resulting reduced circumference of the remaining sector.

We give the formula for the volume of a cone of base radius R and lateral surface length r.

We eliminate the variable r using the length constraint, at least write r in terms of L and theta only.

We set the perimeter of the base of the proposed cone (2 Pi R) equal to the available perimeter of the wire formed sector (2 Pi - theta)r.

We eliminate the variable R or at least write R in terms of L and theta only.

We now substitute into the ConeVol formula for r and R our expressions which involve theta only to obtain the volume of the cone in terms of theta alone.

We specify the length of wire we have, 100 cm.

We plot the volume of the cone formula as a function of theta to see is there is a clear maximum which emerges.

To determine the maximum value of this volume function CVol[theta] we take the derivative of the function with respect to theta and set it equal to zero.

We solve for when the derivative of CVol[theta] = 0.

We now determine the parameters (radius and theta) of our formed shape of wire.

And finally we compute the maximum volume for a cone we can make out of paper from the inside of the wire mesh.

We sketch our solution region.