Application Differentiation Problem

[Vigo]

This is kind of a stupid-sounding problem but here it is:

Stadium Popcorm Problem
A piece of heavy stock paper is cut into a circle with a radius of 4''. The paper is cut form one edge to the center and shaped into a cone-shaped holder. What is the maximum volume of the resulting cone?

Volume of a cone: 1/3pi*r^2*h

This is the exact problem my teacher gave my class and she said if you can do this without my help, it will be extra credit. Well no one in my class knows how to do it and I could really use some extra credit so if someone could help me out that would be great. I guess the title is implying that the problem is related to a snack container at a sports stadium.

I know that I have to differentiate the equation for the volume of a cone but I don't know where to go from there.

 

[BerkMath]

I don't know by what method you are expected to arrive at the solution. Have you ever used a Lagrangian? We want to maximize the volume V(r, h)=1/3*pi*r^2*h, subject to the constraint of surface area S(r,h)=pi*r*sqrt(r^2+h^2)<=pi*R^2. Where r=radius of cone, h=height of cone, R=radius of circle, R=4 inches. From its pretty straight foward. Have you used this method before?

 

[Vigo]

No I have never used this method before. Right now, we are on the chapter of critical points, first and second derivative tests, concavity, and minimum and maximum points. I am only in AP Calculus AB, so the solution should be able to be easily reached. My teacher said that this problem will be related to later sections in this chapter.

 

[mathphys]

hint

Well, lagrangian and/or variational principles are super, but there is a much simpler method of solution.
Find equations for the height and the radius of the cone r and h, in terms of the radius of the circle R and the ANGLE t that R makes with the horizontal (or the angle x between h and R)
____ r
| /
|h /
| / R
|/__t_________
Then use them in the volume formula to transform it in one variable formula and use what you have learned of maxima and minima to find the corresponding t, and from it h and r, and V_max ;).
Regards