At 1:00 a.m., a homicide detective found the reading of a corpse's body temperature to be 88 degrees Fahrenheit. One hour later, the body temperature is 83 degrees Fahrenheit. If the body has been in a 74 degree Fahrenheit room since its death, what is the time of death assuming the body was at 98.6 degrees Fahrenheit at its death?

Question

At 1:00 a.m., a homicide detective found the reading of a corpse's body temperature to be 88 degrees Fahrenheit. One hour later, the body temperature is 83 degrees Fahrenheit. If the body has been in a 74 degree Fahrenheit room since its death, what is the time of death assuming the body was at 98.6 degrees Fahrenheit at its death?

Answer 1

From the "Related Questions" below,

Steve answered this question back in July

http://www.jiskha.com/display.cgi?id=1373408502

Is this a question under Newton's Law of Cooling from Calculus?
If so, just apply the formula which is in your text.

Answer 2

To determine the time of death, we can use Newton's Law of Cooling, which states that the rate of change of an object's temperature is proportional to the difference between its temperature and the surrounding temperature.

Let's break down the information given:

1. The initial body temperature (T0) was 98.6 degrees Fahrenheit.
2. One hour later (after 1:00 a.m.), the body temperature (T1) was 83 degrees Fahrenheit.
3. The surrounding room temperature (Tr) was 74 degrees Fahrenheit.

Using Newton's Law of Cooling, we can set up the following equation:

(T1 - Tr) = (T0 - Tr) * e^(-k * time)

Where:
- e is Euler's number (approximately 2.71828),
- k is the cooling constant (which we need to find),
- time is the duration between time of death and the recorded temperature (which we want to determine).

To find the cooling constant (k), we need two sets of temperature data. We can use the initial body temperature and the reading after one hour:

(T1 - Tr) = (T0 - Tr) * e^(-k * 1)

Let's solve for k:

T1 - Tr = (T0 - Tr) * e^(-k)

Simplifying the equation:

(T1 - Tr) / (T0 - Tr) = e^(-k)

Using natural logarithm (ln) on both sides to isolate k:

ln[(T1 - Tr) / (T0 - Tr)] = -k

Now, let's calculate k:

k = -ln[(T1 - Tr) / (T0 - Tr)]

Once we have the value of k, we can rearrange the formula to find the time (in hours) between the time of death and the recorded temperature:

Time = (1/k) * ln[(T0 - Tr) / (T - Tr)]

Now we can substitute the given values to find the approximate time of death.

From the "Related Questions" below,

16

To determine the time of death, we can use Newton's Law of Cooling, which states that the rate of change of an object's temperature is proportional to the difference between its temperature and the surrounding temperature.