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Thread: Exponential Problem

  1. #1
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    Exponential Problem

    Instant coffee is made by adding boiling water ($\displaystyle 212F$) to coffee mix. If the air temperature is $\displaystyle 70F$, Newton's law of cooling says that after $\displaystyle t$ minutes, the temperature of the coffee will be given by a function of the form $\displaystyle f(t) = 70 - Ae^{-kt}$. After cooling for 2 minutes, the coffee is still $\displaystyle 15F $ too hot to drink, but 2 minutes later it is just right. What is this ideal temperature for drinking?

    $\displaystyle 212 = 70 - Ae^{0}$

    $\displaystyle A = 212 - 70$

    $\displaystyle A = 142$

    $\displaystyle 15 = 70 - 142e^{-k2}$

    $\displaystyle ln{\frac{\frac{(15-70)}{-142}}{-2}} = k$

    $\displaystyle 0.47 = k$

    $\displaystyle f(4) = 70 - 142e^{-0.47\times4}$

    $\displaystyle f(4) = 48.697F$

    ...is my answer correct?
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Macleef View Post
    Instant coffee is made by adding boiling water ($\displaystyle 212F$) to coffee mix. If the air temperature is $\displaystyle 70F$, Newton's law of cooling says that after $\displaystyle t$ minutes, the temperature of the coffee will be given by a function of the form $\displaystyle f(t) = 70 - Ae^{-kt}$. After cooling for 2 minutes, the coffee is still $\displaystyle 15F $ too hot to drink, but 2 minutes later it is just right. What is this ideal temperature for drinking?

    $\displaystyle 212 = 70 - Ae^{0}$

    $\displaystyle A = {\color{red}212 - 70}$

    $\displaystyle A = 142$

    $\displaystyle 15 = 70 - 142e^{-k2}$

    $\displaystyle ln{\frac{\frac{(15-70)}{-142}}{-2}} = k$

    $\displaystyle 0.47 = k$

    $\displaystyle f(4) = 70 - 142e^{-0.47\times4}$

    $\displaystyle f(4) = 48.697F$

    ...is my answer correct?
    49 degrees Fahrenheit seems a bit too cold for a cup of coffee!! XD

    Your mistake is in red. It should be $\displaystyle A=70-212=-142$. That would mean then that $\displaystyle f\left(t\right)=70+142e^{-kt}$

    Try the problem from here. You will have to find a new k value before you can get to the desirable answer!
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  3. #3
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    haha that's what I thought too... thanks! I dislike making careless mistakes
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