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Optimizating problem

  1. Feb 24, 2012 #1
    A chemical manufacturing plant can produce z units of chemical Z given p units of chemical P and r units of chemical R, where:

    z=90 p^.5 r^0.5

    Chemical P costs $400 a unit and chemical R costs $3,200 a unit. The company wants to produce as many units of chemical Z as possible with a total budget of $320,000.

    First thing I did was find the constraint being 400p+3200r=320,000, then solved for p=800-8r
    plugged that into the first eqn z=90(800-8r)^.5*r^.5
    Then I got z=1800(2r)^.5-720r, took the derivative z'=(1800/(2r^.5))-720 set it equal to zero and solved for r. I got r=25/4
    then plugged that into the eqn to solve for p, 400p+3200(25/4)=320,000, p=758.
    Plugged both of those into the original eqn to find z=90(758)^.5*(25/4)^.5=6194.65...
    Somewhere I'm wrong, not quite sure any guidance would help. Thanks
     
  2. jcsd
  3. Feb 24, 2012 #2

    Ray Vickson

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    Your equation [itex] z = 90\sqrt{r} \sqrt{800-8r}[/itex] is correct, but your equation [itex] z = 1800 \sqrt{2r}-720r[/itex] is incorrect. I cannot imagine how how got the second equation, or why you would think it is valid. Certainly, you don't need it: the first equation is perfectly OK for finding the optimum. Although you say you are not quite sure any guidance would help, I am offering it anyway.

    RGV
     
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