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Exponential growth word problem, pie.

  1. Feb 25, 2013 #1
    This whole chapter has been tripping me. My professor made-up another way of doing this, which I'm having a hard time understanding. He's a really intelligent guy(even other professors say it) so he can do this, but I'm having a hard time understanding his way. I went to tutoring today, they couldn't help me with his way. So I'll just try the standard way and see if he'll let it pass.

    How would I normally do this? I know that the standard formula is f(t)=Aert?

    Would it start like f(2)=150er2? That 70 degrees is my biggest problem. I know that it can't go under 70 degrees, I'm having a hard time knowing how that'll fit into this equation. Thank you in advance.
     

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  3. Feb 25, 2013 #2

    SammyS

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    attachment.php?attachmentid=56108&d=1361843020.png

    The temperature of the pie [STRIKE]cools[/STRIKE] decreases exponentially from whatever temperature it had coming out of the oven to room temperature (70° F). You will need a somewhat different fuction than what you showed.

    If f(t) represents the temperature of the pie and t is number of hours after leaving the oven, then:
    f(2) = 150°F

    f(5) = 130°F​
     
    Last edited: Feb 25, 2013
  4. Feb 25, 2013 #3

    eumyang

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    No, it wouldn't. 150°F is not the initial temperature of the pie. It's the temperature after 2 hours. What you would have is [itex]150 = Ae^{2r}[/itex].

    Are you sure you're supposed to use the formula f(t)=Aert? The way I remember learning it, this is Newton's Law of Cooling, and here was the formula:
    [itex]T(t) = T_m + (T_0 - T_m)e^{-kt}[/itex],
    where
    Tm is the temperature of the surrounding medium, and
    T0 is the initial temperature of the object.


    EDIT: Beaten to it. ;)
     
    Last edited: Feb 25, 2013
  5. Feb 25, 2013 #4
    A decaying one? Such as f(t)=Ae-rt? I got this from online. I got that original formula from online. I wasn't too sure of how it worked. Since it allowed me to take the natural log when solving for time on simpler problems, I used it.
     
  6. Feb 25, 2013 #5
    I'm not too sure. Since the chapter is on exponential growth I think that he gave one general formula of some sort to cover all types of exponential growth(interest, bacteria etc..)
     
  7. Feb 25, 2013 #6
    This is not a problem in exponential growth, it is a problem in exponential decay. The thing that is decaying is the difference between the temperature of the pie, and the temperature of the air: (T - 70)
    So,

    time T T-70
    2 150 80
    5 130 60

    For exponential decay, the constant r in your equation is negative.
     
  8. Feb 25, 2013 #7

    SammyS

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    I would suggest modifying your temperature function. Try [itex]\displaystyle \ \ f(t) - 70 = Ae^{-rt}\,, \ [/itex] where f(t) is the temperature of the pie, and t is time in hours, after the pie has been removed from the oven .

    You will need to use logarithms to solve this.
     
  9. Feb 26, 2013 #8
    Using
    the only things you do not know are ##k## and ##T_0##. You can then insert the data you have about temperatures and times and you have a system with two equations and two variables (i.e. ##k## and ##T_0##). Solve the system and you will have your solution with all parameters.

    Finally you just need to insert 7 (hours) and compute the result.
     
    Last edited: Feb 26, 2013
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