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Homework Help: Optimizing Cone - Calculus

  1. Jan 17, 2012 #1
    Hello there :) I'm having tons of trouble figuring out how to finish this problem.
    A cone is to be constructed having a given slant height of l>0 . Find the radius and height which give maximal volume.

    I am unsure of which variables to keep in order for it to be maximized, and how to go about optimizing it.

    This is how I was going about it: I think that the cross-section of the cone makes a right angled triangle, for which the equation would be l^2= b^2 + h^2, and in order to maximize the volume you must relate it to the volume equation V = 1/3(pi)r^2h, but I am having trouble putting it together, to be able to differentiate and then maximize.
  2. jcsd
  3. Jan 17, 2012 #2


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    ok so colume as a function of r & h is
    V(r,h) = 1/3(pi)r^2h

    but you also know (assuming b=r)

    rearranging the contsrtaint gives
    r^2 = h^2-l^2

    and you can subsitute into you volume equation, to get V(h) only. Then you can differentiate w.r.t. h and maximise remembering that l is constant
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