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Differential equations and law of cooling

  1. Oct 27, 2013 #1
    1. The problem statement, all variables and given/known data
    A homicide victim is found to have a temperature of 31°C at the stroke of midnight. At 1:00AM his temperature dropped to 29°C. Assuming that the temperature of the room stays at 20°C, when did the murder take place?

    2. Relevant equations
    -


    3. The attempt at a solution
    This is more of a problem in how to solve for the constants and variables instead of setting up the problem. By noticing that:

    dT/dt = -k(T-N) (where N is the lowest temperature value and T the temperature)

    We obtain a solution to the differential equation which is shown below:

    T(t) = 20 + (T0 - 20)e-kt

    And with this result, we can label the time 12.00AM as h and 1.00AM as h+1:

    T(h) = 20 + (T0-20)e-kt = 31
    T(h+1) = 20 + (T0-20)e-k(t+1) = 29

    With clearing these two expressions (dividing them), we obtain that the value of the constant k is:

    k ≈ 0.2007

    However, now I can't work out anything else with having the value of the constant. I can't clear neither for T0 (the temperature at the beginning) nor k (the constant). I've dealt with a similar problem but in this case the initial temperature was given so the problem could be worked out. However, how can we proceed in this case?
     
  2. jcsd
  3. Oct 28, 2013 #2

    Simon Bridge

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    With the constant k, you now have an equation for the temperature as a function of time.
    What is the temperature of a freshly dead person?
     
  4. Oct 28, 2013 #3
    I'm guessing 37.5 haha. Would this be fair enough? Now i can find To.
     
  5. Oct 28, 2013 #4

    Simon Bridge

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    You could just take a wild guess .... or you could look up "human body temperature".

    IRL: problems do not come with every single value you need written down for you - you have to learn to go look for the them.
     
  6. Oct 28, 2013 #5

    LCKurtz

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    Isn't that pretty close for ##^\circ C##?
     
  7. Oct 29, 2013 #6
    Thank you very much! I never noticed that we could just have assumed the initial value of the temperature (I completely forgot about that fact).
     
  8. Oct 29, 2013 #7

    Simon Bridge

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    ... yes probably - no units were given and nor was the origin of the number provided.

    Human mean body temp is usually quoted as 98.6F or 37.0C.
    http://www.nlm.nih.gov/medlineplus/ency/article/001982.htm
    http://hypertextbook.com/facts/LenaWong.shtml

    This is where you have to show your working to get the marks (or provide a citation if you want to get published.) so I think it is a fair criticism.
    It could just have been pulled out of the air and accidentally come close in some units.

    No worries - technically you should provide a reason for your assumption.
    i.e. if the person died of a fever, you would not assume normal body temp.
    You could also have used the range of body-temps, and ended up with a range of time-of-death values - just like they do on CSI :)
     
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