Finding the maximum volume of a cylinder inscribed in a cone

[theRukus]

Homework Statement

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

Homework Equations

V_cone = (1/3)(pi)(r²)(h)

V_cylinder = (pi)(r²)(h)

The Attempt at a Solution

I've been trying to relate the height of the cylinder to the base of the cylinder. I'm not having much luck. All the equations I make have way too many variables to be optimized..

Could anyone give me a nudge in the right direction?

 

[Mark44]

Homework Statement

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

Homework Equations

V_cone = (1/3)(pi)(r²)(h)

V_cylinder = (pi)(r²)(h)

The Attempt at a Solution

I've been trying to relate the height of the cylinder to the base of the cylinder. I'm not having much luck. All the equations I make have way too many variables to be optimized..

Could anyone give me a nudge in the right direction?

Draw a 2-d sketch of the vertical cross-section of the cone (should look like a triangle). If the center of the base is at the origin, the tip of the cone is at (0, h), and the righthand corner is at (r, 0). You should be able to get the equation of the line between these two points.