dT/dt=� k(T �- Tm), t >� 0
where k is a constant of proportionality, T and Tm are measured in degrees Fahrenheit,
and time t is measured in hours. Because Diff wants to investigate the past using positive
values of time, he decides to correspond t � 0 with 6:00 A.M., and so, for example,
t � 4 is 2:00 A.M. After a few scratches on his yellow pad, Diff realizes that with this time convention the constant k in (1) will turn out to be positive. Diff jots a reminder to himself that 6:30 A.M. is now t=-1/2.
After much deep thought Diff decides to begin by assuming that Mr. Wood was murdered inside the refrigerator. What is the estimated time of death?
To estimate the time of death, we need to solve the given differential equation:
dT/dt = k(T - Tm)
We can solve this using separation of variables.
First, we rearrange the equation:
dT/(T - Tm) = k dt
Now, we integrate both sides of the equation. Let's assume that the initial temperature T(0) = T0.
∫ dT/(T - Tm) = ∫ k dt
ln|T - Tm| = kt + C
where C is the constant of integration.
Using the initial condition T(0) = T0, we can substitute this value into the equation:
ln|T0 - Tm| = 0 + C
C can be obtained by evaluating the natural logarithm:
C = ln|T0 - Tm|
Substituting C into the equation, we have:
ln|T - Tm| = kt + ln|T0 - Tm|
Next, we exponentiate both sides:
|T - Tm| = e^(kt + ln|T0 - Tm|)
Simplifying further:
|T - Tm| = e^(ln|T0 - Tm|)e^(kt)
|T - Tm| = |T0 - Tm|e^(kt)
Now we can separate the absolute values into two cases:
1. T - Tm = (T0 - Tm)e^(kt)
2. T - Tm = -(T0 - Tm)e^(kt)
For case 1, solving for T will give us:
T = Tm + (T0 - Tm)e^(kt)
For case 2, solving for T will give us:
T = Tm - (T0 - Tm)e^(kt)
Now we need to determine the time t when the temperature T equals the temperature inside the refrigerator Tm.
Case 1:
T = Tm + (T0 - Tm)e^(kt)
Tm = Tm + (T0 - Tm)e^(kt)
Simplifying the equation:
0 = (T0 - Tm)e^(kt)
Since k is positive, e^(kt) will never be zero. Therefore, there is no solution for case 1.
For case 2:
T = Tm - (T0 - Tm)e^(kt)
Tm = Tm - (T0 - Tm)e^(kt)
Simplifying the equation:
0 = -(T0 - Tm)e^(kt)
Again, since k is positive, e^(kt) will never be zero. Therefore, there is no solution for case 2.
Based on the analysis, we cannot estimate the time of death assuming Mr. Wood was murdered inside the refrigerator using the given differential equation.