# Homework Help: Optimization largest possible volume problem

1. Nov 16, 2009

### pynergee

1. The problem statement, all variables and given/known data
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

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

3. 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

2. Nov 17, 2009

### lanedance

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

3. Nov 17, 2009

### pynergee

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

4. Nov 17, 2009

### 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