<p>Given the rate of change of temperature of an object is directly proportional

Question

<p>Given the rate of change of temperature of an object is directly proportional to the difference in temperature of the object and the ambient temperature. A murdered victim body's core temperature is measured to be 45 degrees Celsius and is remeasured 30 minutes later and found to be 48 degrees Celsius. Given the temperature in the drying oven where the body was found had been a constant 60 degrees Celsius for many hours and that a living person usually has a body temperature of 37 degrees Celsius , and if evidence suggested that the site of the body was also the site of the crime, how long had the body been in the drying oven before the first temperature measurement?</p>

Explanation / Answer

The model is:

dT/dt=kT=k(T-Tout)

dT/(T-Tout)=kdt

ln(T-Tout)=kt+C1

T-Tout=ekt+C1=Cekt

T(t)=Tout+Cekt

Tout=60

T(o)=60+Ce0=60+C=37 --> C=-23

T(t0)=60-23ekt0=45 --> ekt0=15/23

T(t0+30)=60-23ek(t0+30)=60-23ekt0*e30k=60-23*(15/23)e30k=60-15e30k=48 --> e30k=12/15=0.8

30k=ln(0.8)

k=ln(0.8)/30=-ln(1.25)/30<0

ekt0=e-(ln(1.25)/30)t0=15/23

(-ln(1.25)/30)*t0=ln(15/23)

t0=-30*(ln(15/23)/ln(1.25))57.47[min]

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