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 using Newton's Law of Cooling, we need to find the value of "t" in the equation u(t) = T + (u - T)e^(kt), where u(t) is the body temperature at time t, T is the constant temperature of the surrounding medium, u is the initial temperature of the body, k is a negative constant, and e is the base of the natural logarithm.
Given:
- Initial temperature (u): The body temperature at the time of death is unknown.
- Surrounding temperature (T): This information is not given in the question.
- Temperature at 9:00 AM: 84.6 degrees Fahrenheit
- k for a human body: -1/-2.5
Since we are missing the surrounding temperature (T) and the initial temperature (u), we would need more information to calculate the exact time of death. However, we can still determine an approximate range for the time of death.
Let's assume a typical human body temperature of about 98.6 degrees Fahrenheit as the initial temperature (u) and try to solve for time (t).
u(t) = T + (u - T)e^(kt)
Substituting the given values:
84.6 = T + (98.6 - T)e^((-1/-2.5)t)
Now, we need to solve for "t" by isolating it on one side of the equation.
Simplifying the equation further, we have:
84.6 - T = (98.6 - T)e^((-1/-2.5)t)
Dividing both sides by (98.6 - T):
(84.6 - T) / (98.6 - T) = e^((-1/-2.5)t)
Now, we take the natural logarithm (ln) of both sides to eliminate the exponential term:
ln((84.6 - T) / (98.6 - T)) = (-1/-2.5)t
Simplifying the equation:
t = (-2.5 / -1) * ln((84.6 - T) / (98.6 - T))
To estimate the time of death, we need to know the surrounding temperature (T) and substitute it into the equation. Without that information, it is not possible to calculate the exact time of death.