The problem involves maximizing the 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 as V = πr²h, where h can be derived from the relationship h = 10(1 - r/3) using similar triangles. By substituting this expression for h into the volume formula, the volume becomes V = 10π(r² - r³/3). To find the maximum volume, differentiate this equation with respect to r and set the derivative equal to zero.
PREREQUISITESStudents studying calculus, particularly those focusing on optimization problems, as well as educators teaching geometric relationships and volume calculations.
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.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
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.