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Math Help - Exponential Growth and Logarithms

  1. #1
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    Exponential Growth and Logarithms

    So, I have these two problems for Calculus that I'm rather stuck on right now.

    The information given for the first is this:

    (Exponential growth)
    Colony weight:
    15 mg at 9:00 A.M.
    25 mg at 11:30 A.M.
    45 mg at ?

    where 9:00 A.M. corresponds to t=0.

    What's given for the next one is this:

    "In the cold storage room, which is maintained at a constant temperature of 20° C, Detective Marino discovered the body of Dr. Expo’s assistant, Nat Log. The Detective measured the temperature of the body to be 34° C at 5:00 p.m. Two hours later, when Dr. Kay Scarpetta, the chief medical examiner, arrived to examine the bodies and evidence, she measured Nat’s body temperature to be 31.7° C.
    Dr. Scarpetta knew that Newton’s Law of Cooling says that the temperature of a cooling object is given by the function
    T
    (t)=Sekt+M

    where M is the temperature of the surrounding environment, although she didn’t know either S or k. She did know that the normal body temperature of a living person is 37° C. "

    And I'm supposed to find the missing times for both. Any help would be greatly appreciated!
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  2. #2
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    Quote Originally Posted by Rumor View Post
    The information given for the first is this:

    (Exponential growth)
    Colony weight:
    15 mg at 9:00 A.M.
    25 mg at 11:30 A.M.
    45 mg at ?

    where 9:00 A.M. corresponds to t=0.
    you should already be acquainted with the general equation for uninhibited exponential growth ...

    y = y_0 e^{kt}

    for your first problem ...

    y_0 = 15 mg

    y = 15e^{kt}

    at t = 2.5 hrs, y = 25 mg

    25 = 15e^{2.5k}

    solve for k ... then use the completed equation to find the time when y = 45 mg
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  3. #3
    MHF Contributor Calculus26's Avatar
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    1.

    A(t) = A(0) e^(kt)

    A(0) =15

    A(t) = 15 e^(kt)

    A(2.5) = 25 =15 e^(2.5k)

    5/3 = e^(2.5k)

    ln(5/3) = 2.5 k

    solve for k and you'll Have A(t)

    Set A(t) = 45 and solve for t with ln

    2. T(t) = se^(-kt) + 20

    T(0) = 34 use this to solve for s

    T(2) =31.7 use this to solve for k

    You now have T(t) set this equal to 37 to determinine time of death

    t will be negative the time before the body was first found
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  4. #4
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    Many thanks to the both of you!
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