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Math Help - Exponential function about determing the time of death.

  1. #1
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    Exponential function about determing the time of death.

    Using Newton's Law of Cooling, I have to predict the time of death of a person.

    H-R = Be^-kt is the relationship between the temperature of the body (H) and the constant room temperature (R), t is the time in hours since the body expired and k and B are the constants. Using that equation I have to determine the time of death.

    When the murder is committed, the body, originally at 37'C, cools according to Newton's Law of Cooling, if the body is found at 9:30am the temperature of the body is 35'C and that temperature of the body is 30'C at 4:00pm. Assuming that the temperature of the surrounding air is a constant 20'C.

    I know how to solve simpler exponential models, but I'm confused how to do this one.
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  2. #2
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    Quote Originally Posted by eriiin View Post
    Using Newton's Law of Cooling, I have to predict the time of death of a person.

    H-R = Be^-kt is the relationship between the temperature of the body (H) and the constant room temperature (R), t is the time in hours since the body expired and k and B are the constants. Using that equation I have to determine the time of death.

    When the murder is committed, the body, originally at 37'C, cools according to Newton's Law of Cooling, if the body is found at 9:30am the temperature of the body is 35'C and that temperature of the body is 30'C at 4:00pm. Assuming that the temperature of the surrounding air is a constant 20'C.

    I know how to solve simpler exponential models, but I'm confused how to do this one.
    At t_0=0 we have:

    H-R=17=Be^0=B

    So B=17.

    At t=9.5 (we are working in decimal hours here):

    H-R=15=17e^{-k(9.5-t_0)}

    At t=16.0 (4 pm is 16.00hrs):

    H-R=10=17e^{-k(16.0-t_0)}

    Now take logs and solve the resulting equations for $$ t_0 and $$ k

    CB
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  3. #3
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    I'm a little confused, what does the o in to mean? And why is it 17d^-k(9.5-t)?
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  4. #4
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    Quote Originally Posted by eriiin View Post
    I'm a little confused, what does the o in to mean? And why is it 17d^-k(9.5-t)?
    $$ t_0 denotes the time of death

    The second question I don't understand, $$ 9.5 hours is 09:30 so 9.5-t_0 is the time since death at 09:30 in hours

    CB
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  5. #5
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    Sorry I should have been more specific, I just don't understand why it's 9.5 - t, like why is that we subtract it?
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  6. #6
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    Quote Originally Posted by eriiin View Post
    Sorry I should have been more specific, I just don't understand why it's 9.5 - t, like why is that we subtract it?
    It is the time from death at 09:30

    CB
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