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Optimization largest possible volume problem

  1. Nov 16, 2009 #1
    1. The problem statement, all variables and given/known data
    A right circular cylinder is inscribed in a cone with height h and base radius r. Find the largest possible volume of such a cylinder.

    Volume of a cylinder = (pi)(r^2)h
    Volume of a cone = (1/3)(pi)(r^2)h


    2. Relevant equations
    Volume of a cylinder = (pi)(r^2)h
    Volume of a cone = (1/3)(pi)(r^2)h



    3. The attempt at a solution
    Technically this is my roommate's problem, he is in Calc1. He has been having some problems with this one, and I'm in Differential Equations, and I can't remember how to really do this one. I know you want to find the derivative of the cylinder, and find when it is equal to zero, but I am stumped on how to approach the problem
     
  2. jcsd
  3. Nov 17, 2009 #2

    lanedance

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    solve for the height of the cylinder in terms of its radius then look at optimisation through substitution or lagrange multipliers
     
  4. Nov 17, 2009 #3
    I know that, but how? Would you use similar triangles or something like that?
     
  5. Nov 17, 2009 #4

    lanedance

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    you could do that...

    draw a vertical slice of the cone thorough its centre. The cone appears as an isoceles triangle, whilst the cylinder is a rectangle inscribed in the triangle. Use some trig to derive the relation between cylinder height & base
     
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