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Quotient rule with additional condition

  1. May 8, 2013 #1
    1. The problem statement, all variables and given/known data

    I am working on chemical reaction engineering problem and it involves some math, which I am not able to figure out...

    I have to find the residence time for maximum production, which is in the case when : (dη_p)/dτ=0

    I have to find the τ (residence time).

    2. Relevant equations

    η_p=0.1τ/(1+0.4τ+0.04τ^2)

    3. The attempt at a solution

    I figure I have to use the quotient rule:

    d[f/g]=(f'g-fg')/g^2

    f=0.1τ
    g=1+0.4τ+0.04τ^2

    When I use the rule, I come up with an number which is not the same as in the answers. The reason is I think that I don't know how to take into account the condition that (dη_p)/dτ=0
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. May 8, 2013 #2

    ehild

    User Avatar
    Homework Helper
    Gold Member

    The derivative has to be zero, that means an equation for τ, easy to solve. Show your further work please.

    ehild
     
  4. May 8, 2013 #3
    When you say the derivative equals zero, do you mean this:

    0=(f'g-fg')/g^2

    0=[0.1(1+0.4τ+0.04τ^2)-(0.1τ)(0.4+0.08τ)]/g^2
    0=0.1-0.004τ^2/g^2

    0.1-0.004τ^2=0

    and solving for τ=5 , which is the correct answer.

    However, I don't understand why g^2=0 ?
     
  5. May 8, 2013 #4

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    g^2 is NOT 0. What makes you think it is?
     
  6. May 8, 2013 #5
    I don't have any scientific reasoning behind my logic, but basically that is the only way I could see that I come up with the answer given in the book.


    If I subsitute g^2=(1+0.4τ+0.04τ^2)^2 and using the formula
    (a+b+c)^2= a^2+b^2+c^2+ 2(ab+bc+ca)

    I end up with

    0=(f'g-fg')/g^2=(0.1-0.004τ^2)/(1+0.8τ+0.24τ^2+0.032τ^3+0.0016τ^4)

    What would be the next step from here?
     
  7. May 8, 2013 #6

    SteamKing

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    Staff Emeritus
    Science Advisor
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    If g^2 = 0, the derivative is undefined. The quantity (f'g - fg') must be 0 to satisfy the equation.
     
  8. May 8, 2013 #7

    Mark44

    Staff: Mentor

    To elaborate on what SteamKing said, the basic idea is that if a/b = 0, then a = 0. Of course, b cannot be 0.
     
  9. May 9, 2013 #8
    Thanks everyone!
     
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