Optimization Problem

[spar134]

I'm completely stuck on this problem and have no idea how to go about solving it. The problem is: A cone has a slant height of L. What measure must the angle between the side of the cone and the base have in order to produce the largest possible volume? (Remember L is a constant.)

I'm assuming the equations involved will be V=(1/3)(pi)r^2h
I also know if you draw the height in a cone, you produce a right triangle with the slant height. I made the equations cosx=r/L with r standing for radius and tanx=h/r or h=rtanx, but I don't know what else to do. Any help would be appreciated. Thank you!

 

[Bob Brown MSEE]

I'm completely stuck on this problem and have no idea how to go about solving it. The problem is: A cone has a slant height of L. What measure must the angle between the side of the cone and the base have in order to produce the largest possible volume? (Remember L is a constant.)

I'm assuming the equations involved will be V=(1/3)(pi)r^2h
I also know if you draw the height in a cone, you produce a right triangle with the slant height. I made the equations cosx=r/L with r standing for radius and tanx=h/r or h=rtanx, but I don't know what else to do. Any help would be appreciated. Thank you!

You can get V(r) by letting:
h = SQRT(L^2 - r^2)

then find r(L) that maximizes V(r)
Hint: Don't worry about the angle until you get r(L)