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Homework Help: Optimization problem

  1. May 3, 2010 #1
    1. The problem statement, all variables and given/known data

    A right circular cone of base radius r and height h has a total surface area S and volume V . Show that 9V2=r2(S2-2pir2S) . (i can do this part) . Hence or otherwise , show that for a fixed surface area S , the maximum volume of the cone occurs when its semi-vertical angle , theta is given by tan theta=1/2(root 2)

    3. The attempt at a solution

    From the proven equation ,


    Differentiate this wrt to r ,

    dV/dr=(2S2r-8pi Sr3)/(18V)

    dV/dr=0 , S=4pi r2 , substitue S with the area of cone , then
    tan theta=r/h=1/(2 root 2)

    This is my question , how do i prove that its a maximum ?
  2. jcsd
  3. May 3, 2010 #2
    Re: optimization

    when you took the derivative of the Volume function and set it equal to zero you are finding a critical value.
    There are a couple ways to test whether it is a local min or max. The second derivative test is one of them.

    note: I have to check your derivation; not that I am doubting it or anything.
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