Caculus Question that involves volume

So the question is basicly summed up in this picture I drew, my teacher told me we have to use composite function and take the derivitive of the final equation to find the small r and H that will maximize volume
http://imageshack.us/photo/my-images/15/calcquestion.png/

We want the largest cone to be inscribed in a sphere of radius r, where the radius of the cone is R.
So you are looking for R and H where r is the radius of the sphere.
V(cone) = (1/3) pi*R^2* H

H and R are variables, but r is a fixed radius, so it is constant. We need V in terms of one variable.

Draw a radius from the center of the sphere to the outer edge of the base of the cone.
You should have a right triangle: R^2 +(H-r)^2= r^2

R^2 + H^2-2Hr+ r^2= r^2

R^2 = 2Hr-H^2

So V = (pi/3) (2Hr-H^2)*H

V= (pi/3) (2rH^2- H^3)

V' = dV/dH= (pi/3) ( 4rH- 3H^2)= 0

(pi/3)(H)(4r- 3H)= 0

H= 4r/3

Then find R:

R^2= r^2- (4r/3-r)^2

R^2= r^2 -(r/3)^2

= r^2-r^2/9
= 8r^2/9

R= 2rsqr(2)/3

Plug R and H in to find the volume of the cone, if needed.

Hoping this helps!