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Question

integral calculus:

Q: in a murder investigation, the crime scene investigators arrived at 5 a.m to examine the body. They found the body temperature to be 30 degrees celcius in a constant room temperature of 20 degrees celcius. (normal body temperature is taken to be 37 degrees celcius). At 5.30 a.m the coroner arrived and measured the body temperature to be 28 degrees celcius and estimated the time of death to be t hundred hours.
Find t using newtons law of cooling, which states that the rate of cooling at any instant is directly proportional to the difference in temperature between the object and its surroundings, i.e.

dT/dt = k[ T(subscript o) - T (subscript s) ]

Answer 1

Newton's law of cooling should be
dT/dt = k[T(sub s) - T]
since if T>T(sub s), then T decreases with t, where T(sub s) is T of surroundings.

Rewrite it as
dT/dt +kt = kT(sub s)
The solution of this differential equation is
T = Cexp(-kt) + T(sub s)
where C is a constant of integration.
From the given conditions
30 = Cexp(-5k) + 20
28 = Cexp(-5.5k) + 20
37 = Cexp(-kt') + 20
where t' is time of death in hours clock time.
Solve the first two eauations for
k = -2ln(0.8) = .44629 (ln is natural log)
C =93.13336
and by the third equation
17/C = exp(-kt')
-kt' = ln(17/C)
t' = 3.811 hours clock time