A dead body was found within a closed room of a house where the temperature was a constant 70° F. At the time of discovery the core temperature of the body was determined to be 85° F. One hour later a second measurement showed that the core temperature of the body was 80° F. Assume that the time of death corresponds to t_0 and that the core temperature at that time was 98.6° F. Determine how many hours elapsed before the body was found. Can you solve this using T=Tm+(To-Tm)e^kt

This problem involves using **Newton's Law of Cooling** to determine how long it has been since the person died. The general formula for Newton's Law of Cooling is:

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

Where:
- \(T(t)\) is the temperature of the object (the body) at time \(t\),
- \(T_m\) is the ambient temperature (the surrounding temperature, which is constant at 70°F),
- \(T_0\) is the initial temperature of the body at time \(t = 0\), which is 98.6°F (normal human body temperature),
- \(k\) is a constant related to the cooling rate,
- \(t\) is the time elapsed since the body started cooling,
- \(e\) is the base of the natural logarithm.

### Step 1: Setup the known information

From the problem, we know:
- The ambient temperature \(T_m = 70^\circ F\),
- The initial temperature of the body \(T_0 = 98.6^\circ F\),
- The temperature at the time of discovery (\(t_f\)) is \(85^\circ F\),
- One hour later, the temperature is \(80^\circ F\).

We want to find the time \(t_f\) that has passed since death when the body was discovered (i.e., the time when the body reached 85°F).

### Step 2: Write equations using the data

At \( t = t_f \), the body temperature was \( T(t_f) = 85^\circ F \). Using the formula, we can write:

\[
85 = 70 + (98.6 - 70)e^{kt_f}
\]

At \( t = t_f + 1 \) (one hour later), the temperature was \( T(t_f + 1) = 80^\circ F \). This gives another equation:

\[
80 = 70 + (98.6 - 70)e^{k(t_f + 1)}
\]

Now we have two equations with two unknowns: \( t_f \) and \( k \). We can solve these to find the time \( t_f \).

### Step 3: Solve the system of equations

We will solve these equations to find \( t_f \) and \( k \). Let's do that now.The solution to the system of equations gives the following values:
- \( k = -0.405 \) (rate constant),
- \( t_f = 1.59 \) hours.

This means that approximately **1.59 hours** had elapsed before the body was found. Therefore, the time of death occurred about 1.59 hours before the body temperature was measured at 85°F.

Would you like more details on any part of this solution? Here are five questions related to the solution that could deepen your understanding:

1. How is Newton's Law of Cooling derived?
2. How does the rate constant \(k\) influence the cooling process?
3. What assumptions are made in applying Newton's Law of Cooling in real-world scenarios?
4. How would the solution change if the ambient temperature fluctuated?
5. How could you estimate the uncertainty in the time of death using this model?

**Tip**: For more accurate results in forensic science, it's important to account for factors like body size, clothing, and environmental conditions that may affect the rate of cooling.

A dead body was found within a closed room of a house where the temperature was a constant 70°F. At the time of discovery, the core temperature of the body was determined to be 85°F. One hour later, a second measurement showed that the core temperature of the body was 80°F. Assume that the time of death corresponds to t_0 and that the core temperature at that time was 98.6°F. Determine how many hours elapsed before the body was found. Can you solve this using T=Tm+(To-Tm)e^kt?

Solve a forensic problem using Newton's Law of Cooling. Given a body found in a room at 70°F with initial and subsequent temperature measurements, this problem guides you through calculating the time of death. Learn about exponential decay and how differential equations are applied in real-world scenarios.

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