How to Optimize Cone Volume with Given Slant Height?

[girlygirl93]

Hello there :) I'm having tons of trouble figuring out how to finish this problem.
A cone is to be constructed having a given slant height of l>0 . Find the radius and height which give maximal volume.



I am unsure of which variables to keep in order for it to be maximized, and how to go about optimizing it.



This is how I was going about it: I think that the cross-section of the cone makes a right angled triangle, for which the equation would be l^2= b^2 + h^2, and in order to maximize the volume you must relate it to the volume equation V = 1/3(pi)r^2h, but I am having trouble putting it together, to be able to differentiate and then maximize.

 

[lanedance]

ok so colume as a function of r & h is
V(r,h) = 1/3(pi)r^2h

but you also know (assuming b=r)
l^2=h^2+r^2

rearranging the contsrtaint gives
r^2 = h^2-l^2

and you can subsitute into you volume equation, to get V(h) only. Then you can differentiate w.r.t. h and maximise remembering that l is constant