An application of Newtonian Cooling is calculating the time of death of a person. When healthy,...

An application of Newtonian Cooling is calculating the time of death of a person. When healthy, a human body has a steady temperature of 37◦C. Once a person dies, the regulatory mechanisms stop working, and the body temperature rises or falls, depending on the ambient temperature of the environment they died in.

1. The temperature of the body will decrease at a rate proportional to the difference in the current temperature and the ambient temperature. Based on this statement, show how we can model body temperature with the ODE

dT/ dt + kT = kT∞

where T is the temperature of the body, T∞ is the ambient temperate and k is constant.

2. Solve the 1st order ODE for T(t) given the body is initially 37◦C and ambient temperature is constant at 24◦C. Leave k as an unknown.

3. The body takes 2 hours to drop to 32◦C. Use this information to calculate the cooling coefficient k.

4. Plot the temperature of the body for 12 hours after death in MATLAB. Make sure both axes are clearly labelled. Comment on whether the plot is correctly modelling the temperature of the body and provide reasoning as to why (Hint: there are three observations you should be able to comment on!).

5. You find another body of a person related to the first (i.e. same k value), but this was outside where the ambient temperature can be approximated as:

T∞(t) = 6sin( πt /12 ) + 24

Solve the ODE in 1. for T(t) using this function for T∞.

Expert Solution

p

Colby Messinger answered 2 years ago

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