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.

thanks

K= -1/-2.5 sorry for the mix up

To estimate the time of death, we can make use of Newton's Law of Cooling and the given body temperature measurements.

The equation for Newton's Law of Cooling is given as:
u(t) = T + (u - T) * e^(kt)

In this case, T represents the constant temperature of the surrounding medium, u is the initial temperature of the body, t represents the time elapsed since death, and k is a negative constant specific to a human body.

Given information:
- At 9:00 AM, the body temperature was 84.6 degrees Fahrenheit.
- We need to estimate the time of death.

Let's plug this information into the equation and solve for t, the time elapsed since death.

u(t) = 84.6
T = ambient air temperature (not provided, assumed constant for simplicity)
u = initial body temperature (not provided, assumed constant for simplicity)
k = -1/(-2.5)

Thus, the equation becomes:
84.6 = T + (u - T) * e^(-1/(-2.5) * t)

Since we don't know T or u, we can simplify the equation by assuming that T = u, meaning that both the initial body temperature and the ambient air temperature are equal. With this assumption, the equation becomes:
84.6 = T + (T - T) * e^(-1/(-2.5) * t)
84.6 = T * e^(-1/(-2.5) * t)

Now, we can solve for t, the time elapsed since death.

Divide both sides of the equation by T:
84.6 / T = e^(-1/(-2.5) * t)

Take the natural logarithm (ln) of both sides to isolate t:
ln(84.6 / T) = (-1/(-2.5) * t)

Simplify the equation:
t = -2.5 * ln(84.6 / T)

Since we don't know the exact ambient air temperature (T), we can assume a typical value for indoor ambient air temperature, which is around 70 degrees Fahrenheit.

Let's calculate the estimated time of death by plugging the values into the equation and evaluating it.

t = -2.5 * ln(84.6 / 70)
t ≈ -2.5 * ln(1.2086)
t ≈ -2.5 * 0.1916
t ≈ -0.4791 hours

The negative sign indicates that time elapsed before the body was found, so we need to convert it to positive. Therefore, the estimated time of death is approximately 0.4791 hours, which is equivalent to about 28.75 minutes.

Thus, based on the given body temperature measurement at 9:00 AM, the estimated time of death is around 28.75 minutes before the body temperature was taken.