Word problem

[Guest]

A huge conical tank is to be made from a circular piece of sheet metal of radius 10 meters by cutting out a sector with vertex angle theta and then welding together the straight edges of the remaining piece. Find theta so that the resulting cone has the largest possible volume.

I have no idea of how to do this problem if you could help that'd be great.

 

[galactus]

Here's one way of looking at it.

Start with the volume of a cone formula:

\(\displaystyle \L\\V=\frac{1}{3}{\pi}r^{2}h\)

Since the slant height of the cone is the radius of the circle:

\(\displaystyle \L\\r^{2}+h^{2}=100\)

Here's a trick, so to speak. Compare circumference to arc subtended by theta, using \(\displaystyle s=r{\theta}\)

\(\displaystyle \L\\2{\pi}r=10(2{\pi}-{\theta})\)

\(\displaystyle \L\\V(h)=\frac{{\pi}}{3}(100-h^{2})h=\frac{{\pi}}{3}(100h-h^{3})\)

\(\displaystyle \L\\V'(h)=\frac{{\pi}}{3}(100-3h^{2})\)

Now, set to 0 and solve for h. From there you can find r and theta.