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Maximum Value problem

  1. Oct 20, 2009 #1
    1. The problem statement, all variables and given/known data
    Hello:

    This is a max/min problem. I am asked to find the rectangular box of maximum volume inscribed in a hemisphere of radius R.


    2. Relevant equations
    So I am wondering if I have set up this correctly. If say my length is x, width is y, and height is z. So, I would have max(xyz). And then would I have [tex]R^{2}-\frac{x^{2}}{4}-\frac{y^{2}}{4} = z^{2}[/tex] I am not sure if this is the correct relationship. And then my function would be: f(x, y) = [tex]xy(r^{2}-\frac{x^{2}}{4}-\frac{y^{2}}{4})[/tex]

    I think I can maximize xyz using xyz^2.

    thanks!
     
  2. jcsd
  3. Oct 20, 2009 #2
    Firstly, what method are you aiming to use in order to maximize [tex]xyz[/tex]? You need to be clear in your own mind what you are going to do with f.

    Setting up the problem is usually a matter of defining your constraints. i.e.
    What hemisphere is your constraint [tex] R^{2}-\frac{x^{2}}{4}-\frac{y^{2}}{4} = z^{2} [/tex] describing?
    Are [tex]x,y,z[/tex] all positive or can some be negative?

    Also, before you get going in these problems, do you have a feel for what the correct answer might be?
     
  4. Oct 20, 2009 #3
    it's a hemisphere, I would think that z would be all positive because I am assuming that it's the upper hemisphere. I am going to use either Lagrange multiplier's method or the usual gradient = 0 method. I am sure the answer will be some multiple of R.
     
  5. Oct 20, 2009 #4
    Yes the answer is likely to involve R somewhere. However, your equation for the hemisphere is slightly off: can you write down the equation for a sphere of radius R in terms of x,y and z?
    Once you have that, it is a simple case of restricting z to be positive as you said.

    Also, maximizing the volume, V, is equivalent to maximizing V^2 so to simplify algebra you can take x^2 y^2 z^2 rather than xyz^2 which is not necessarily the same
     
  6. Oct 20, 2009 #5
    Hello:

    Thanks, so would the equation for a sphere be:
    [tex]x^{2} + y^{2} + z^{2} = R^{2}[/tex]
    So then my constraint is [tex]z^{2} = R^{2}-x^{2}-y^{2}?[/tex]
    And then my function would be to maximize:
    [tex]x^{2}y^{2}(R^{2}-x^{2}-y^{2})?[/tex]
    Thanks.
     
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