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Easy optimization

  1. Apr 16, 2007 #1
    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
    [tex]V=\pi*r^2*h[/tex]



    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. jcsd
  3. Apr 16, 2007 #2
    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.
     
  4. Apr 17, 2007 #3
    Calc 1, we are trying to maximize the volume of a cyclinder inside a cone with the given information.
     
  5. Apr 17, 2007 #4

    HallsofIvy

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    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.
     
    Last edited: Apr 17, 2007
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