Optimization, again

[racuna]

A right circular cylinder is inscribed in a cone with the height h and base radius r. Find the largest possible volume of such a cylinder.

I don't even understand the problem
Can someone clarify the whole thing?

 

[Unco]

G'day, Racuna.

A diagram is always useful.

Let R be the radius of the cylinder;
 H be its height.

/|\       |\
 |        | \
 |        |  \
 |h       |___\
 |     /|\| R |\    
 |   H  | |   | \
\|/    \|/|___|__\
          <------>
             r   
From similar triangles:

r      R
- = -------  [1]
h    h - H

The volume of the cylinder is
 
V = pi/3 * R^2 * H [2]

Use [1] to relate the two variables R and H
 and maximise [2] by differentiating and 
 setting to zero, noting that r and h are constants.