Max Volume of Cylinder Inscribed in Cone: 10r^3π

[Weave]

Homework Statement

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.

Homework Equations

$$V=\pi*r^2*h$$

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.

 

[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.

 

[Weave]

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

 

[HallsofIvy]

Homework Statement

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.

Homework Equations

$$V=\pi*r^2*h$$

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.

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.