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

Staff Emeritus
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: Apr 17, 2007