A body was found in the basement of the underwater basket weaving building at 12:00 noon today, where the temperature is steady 65 degrees Fahrenheit. When found, the core temperature was 91.4 degrees Fahrenheit. Two hours later, at 2:00 pm, the core temperature had fallen to 87.1. Assuming the Body temperature was 98.6 at the time of death use Newton's law of cooling to find the time of death. The time of death was about how many hours before the body was found ?

Newton's Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference between the object's temperature and the ambient temperature. Mathematically, this can be written as:

\[
\frac{dT}{dt} = -k(T - T_{\text{env}})
\]

where:
- \(T(t)\) is the temperature of the body at time \(t\),
- \(T_{\text{env}}\) is the ambient temperature (65°F in this case),
- \(k\) is the cooling constant,
- \(t\) is time (in hours),
- \(T_0\) is the initial temperature of the body (98.6°F at the time of death).

### Step-by-step procedure:

We use the formula derived from Newton's Law of Cooling:

\[
T(t) = T_{\text{env}} + (T_0 - T_{\text{env}}) e^{-kt}
\]

Where:
- \(T(t)\) is the temperature of the body at time \(t\),
- \(T_0 = 98.6^\circ F\),
- \(T_{\text{env}} = 65^\circ F\),
- \(T(0)\) is the temperature of the body when it was found (91.4°F),
- \(T(2)\) is the temperature two hours later (87.1°F).

We first need to find \(k\), the cooling constant, and then use that to find the time of death.

### Step 1: Find \(k\)
Using the first condition when the body is found at noon (\(t = 0\)):

\[
T(0) = 65 + (98.6 - 65) e^{-k(0)} = 91.4
\]
\[
91.4 = 65 + 33.6
\]
Simplifying this to find \(e^0 = 1\), which holds true. Now, let us find \(T_2

A body was found in the basement of the underwater basket weaving building at 12:00 noon today, where the temperature is steady 65 degrees Fahrenheit. When found, the core temperature was 91.4 degrees Fahrenheit. Two hours later, at 2:00 pm, the core temperature had fallen to 87.1. Assuming the Body temperature was 98.6 at the time of death use Newton's law of cooling to find the time of death. The time of death was about how many hours before the body was found?

Learn how to apply Newton's Law of Cooling to estimate the time of death in forensic science. This example involves a body found at 91.4°F, with a steady ambient temperature of 65°F, and cooling down to 87.1°F two hours later. Includes step-by-step calculations using exponential decay and differential equations.

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