Optimization largest possible volume problem

[pynergee]

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.

Volume of a cylinder = (pi)(r^2)h
Volume of a cone = (1/3)(pi)(r^2)h

Homework Equations

Volume of a cylinder = (pi)(r^2)h
Volume of a cone = (1/3)(pi)(r^2)h

The Attempt at a Solution

Technically this is my roommate's problem, he is in Calc1. He has been having some problems with this one, and I'm in Differential Equations, and I can't remember how to really do this one. I know you want to find the derivative of the cylinder, and find when it is equal to zero, but I am stumped on how to approach the problem

 

[lanedance]

solve for the height of the cylinder in terms of its radius then look at optimisation through substitution or lagrange multipliers

 

[pynergee]

I know that, but how? Would you use similar triangles or something like that?

 

[lanedance]

you could do that...

draw a vertical slice of the cone thorough its centre. The cone appears as an isoceles triangle, whilst the cylinder is a rectangle inscribed in the triangle. Use some trig to derive the relation between cylinder height & base