The problem involves finding the maximum volume of a right circular cylinder inscribed in a cone with a height of 10 and a base radius of 3. The volume of the cylinder is expressed in terms of its radius and height.
There is an ongoing exploration of the relationships between the dimensions of the cone and cylinder. Some participants have provided insights into the use of similar triangles and differentiation to approach the problem. However, there is no explicit consensus on the best method to proceed.
Participants note the need to clarify the problem's requirements, including the specific focus on maximizing the volume of the cylinder rather than other potential interpretations. There is also mention of the level of calculus involved, indicating a range of approaches being considered.
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 [itex]V= \pi r^2 h[/itex] gives [itex]V= 10\pi (r^2- r^3/3)[/itex]. Differentiate that with respect to r and set the derivative equal to 0.Weave said: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
[tex]V=\pi*r^2*h[/tex]
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.