Newton’s Law of Cooling states that the temperature of a heated object decreases exponentially over time toward the temperature of the surrounding medium. The temperature u of a heated object at a given time t can be modeled by the following function:

u(t)=T+(u-T)e^kt K<0

where T is the constant temperature of the surrounding medium,
u is the initial temperature of the heated object, and
k is a negative constant for a specific object

Our normal body temperature is usually warmer than the ambient air temperature indoors. Therefore, upon death, the body temperature will exponentially decay according to the function above. For the human body, k = -1/-2.5, if temperature is measured in Fahrenheit.

Q) At 8:30 AM, the police forensic team was called to the Calgary home of a person who had apparently been murdered sometime during the night. In order to estimate the time of death, the person’s body temperature was measured. At 9:00 AM, the body temperature was 84.6 degrees Fahrenheit. Estimate the time of death.

To estimate the time of death, we can use Newton's Law of Cooling and the given equations.

We are given that the initial temperature of the body (u) is unknown, the surrounding temperature (T) is also unknown, and the temperature at 9:00 AM (u(30)) is 84.6 degrees Fahrenheit.

Using the equation for Newton's Law of Cooling: u(t) = T + (u - T)e^kt, we can substitute the given values and solve for the time of death (t).

Let's plug in the values we know:
u(t) = 84.6
T = unknown
u = unknown
k = -1/-2.5 = 0.4

So, the equation becomes:
84.6 = T + (u - T)e^(0.4t)

Since we don't know the values of T and u, we have two unknowns. However, we can use the fact that our normal body temperature is usually warmer than the ambient air temperature indoors to estimate T.

According to medical literature, normal human body temperature is around 98.6 degrees Fahrenheit. So, we can assume this is the initial temperature of the body:
u = 98.6

Substituting this value into the equation, we get:
84.6 = T + (98.6 - T)e^(0.4t)

Now, we have only one unknown left, which is the surrounding temperature T.

Since the person was found in their home, we can estimate the indoor temperature to be around 70 degrees Fahrenheit. So, we can assume this as the surrounding temperature T.
T = 70

Substituting this value into the equation, we get:
84.6 = 70 + (98.6 - 70)e^(0.4t)

Now, we have only one unknown left, which is the time of death t. We can solve this equation using logarithms or by using trial and error.

One way to solve it is by using logarithms. Taking the natural logarithm (ln) of both sides, we get:
ln(84.6 - 70) = ln(28.6) = ln(28.6)e^(0.4t)

Now, we can isolate t by dividing both sides by ln(28.6):
0.4t = ln(84.6 - 70) / ln(28.6)

Finally, we can solve for t by dividing both sides by 0.4:
t = ln(84.6 - 70) / (0.4 * ln(28.6))

Using a calculator or software, we can evaluate this expression to find the estimated time of death.