An application of Newtonian Cooling is calculating the time ofdeath of a person. When healthy, a human body has a steadytemperature of 37â—¦C. Once a person dies, the regulatory mechanismsstop working, and the body temperature rises or falls, depending onthe ambient temperature of the environment they died in.

1. The temperature of the body will decrease at a rateproportional to the difference in the current temperature and theambient temperature. Based on this statement, show how we can modelbody temperature with the ODE

dT/ dt + kT = kT∞

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

2. Solve the 1st order ODE for T(t) given the body is initially37â—¦C and ambient temperature is constant at 24â—¦C. Leave k as anunknown.

3. The body takes 2 hours to drop to 32â—¦C. Use this informationto calculate the cooling coefficient k.

4. Plot the temperature of the body for 12 hours after death inMATLAB. Make sure both axes are clearly labelled. Comment onwhether the plot is correctly modelling the temperature of the bodyand provide reasoning as to why (Hint: there are three observationsyou 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 temperaturecan be approximated as:

T∞(t) = 6sin( πt /12 ) + 24

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