# Homework Help: Easy optimization

1. Apr 16, 2007

### Weave

1. The problem statement, all variables and given/known data
A right circular cylinder in inscribed in a cone with height 10 and base radius 3. Find the largest possible voluem of such a cylinder.

2. Relevant equations
$$V=\pi*r^2*h$$

3. The attempt at a solution
Ok, so I used similar triangles of the cone and cylinder to obtain h=(10/3)r
I substituted that in for h and I'm not sure where to go from there.

2. Apr 16, 2007

### Mindscrape

Volume or surface area? What level of calculus is this? Were you trying to give the volume of a cone or of the cylinder? I would solve the problem with calculus of variations by minimizing the integral of the surface area, but that is something that requires differential equations.

3. Apr 17, 2007

### Weave

Calc 1, we are trying to maximize the volume of a cyclinder inside a cone with the given information.

4. Apr 17, 2007

### HallsofIvy

Look more closely at your similar triangles. You have a large triangle (the entire cone) and a small triangle (the area inside the cone above the cylinder). If the cylinder has height h and radius r, then similar triangles gives (10-h)/r= 10/3 or 10-h= (10/3)r so h= 10- (10/3)r= 10(1- r/3). Putting that into $V= \pi r^2 h$ gives $V= 10\pi (r^2- r^3/3)$. Differentiate that with respect to r and set the derivative equal to 0.

Last edited by a moderator: Apr 17, 2007