A body was found in the basement of the Underwater Basket Weaving Building at​ 12:00 noon​ today, where the temperature is a steady 65 degrees Fahrenheit.  When​ found, the core temperature was 91.1 degrees Fahrenheit.  Two hours​ later, at​ 2:00 PM, the core temperature had fallen to 86.9.   Assuming that the body temperature was 98.6 at the time of​ death, use​ Newton's law of cooling to find the time of death.  
ROUND TO 2 DECIMAL PLACES.
The time of death was about 
  
enter your response here hours before the body was found.

We will use **Newton's Law of Cooling** to solve this problem. Newton's Law of Cooling is given by the equation:

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

Where:
- \( T(t) \) is the temperature of the body at time \( t \),
- \( T_{\text{env}} \) is the ambient (environment) temperature (65°F in this case),
- \( T_0 \) is the initial temperature of the body (98.6°F, which is the normal human body temperature),
- \( k \) is the cooling constant,
- \( t \) is the time elapsed since death.

### Step 1: Set up the known conditions

- At 12:00 noon (time \( t = 0 \)), the body temperature \( T(0) = 91.1^\circ F \).
- Two hours later, at 2:00 PM, the body temperature \( T(2) = 86.9^\circ F \).
- The ambient temperature \( T_{\text{env}} = 65^\circ F \).

We need to first find the cooling constant \( k \) using the two known data points (at 0 hours and 2 hours) and then use that to determine the time of death.

### Step 2: Use Newton's law to find \( k \)

We will first solve the equation for \( k \) using the information at 2:00 PM.

1. At \( t = 2 \) hours, we have the temperature \( T(2) = 86.9 \). Plug this into the equation:
   \[
   86.9 = 65 + (91.1 - 65) e^{-2k}
   \]
   Simplifying:
   \[
   86.9 = 65 + 26.1 e^{-2k}
   \]
   Subtracting 65 from both sides:
   \[
   21.9 = 26.1 e^{-2k}
   \]
   Dividing by 26.1:
   \[
   \frac{21.9}{26.1} = e^{-2k}
   \]
   Taking the natural logarithm of both sides:
   \[
   \ln\left(\frac{21.9}{26.1}\right) = -2k
   \]
   Solving for \( k \):
   \[
   k = -\frac{1}{2} \ln\left(\frac{21.9}{26.1}\right)
   \]

### Step 3: Calculate \( k \) and find the time of death

Let's calculate \( k \), and then use it to determine how long before noon the death occurred.The cooling constant \( k \) is approximately \( 0.0877 \). 

Using this value, we find that the time of death was approximately **2.88 hours** before the body was found at 12:00 noon.

Thus, the time of death was around **9:07 AM**.

Would you like further details on the calculations or any related explanations? Here are some related questions:

1. What assumptions does Newton's law of cooling make in this scenario?
2. How would different ambient temperatures affect the time of death calculation?
3. What are other real-life applications of Newton's law of cooling?
4. How accurate is the estimation of time of death using body temperature?
5. What factors could affect the cooling rate of a body?

**Tip:** Always consider environmental factors like humidity or insulation, which can affect cooling rates in real-world scenarios.

A body was found in the basement of the Underwater Basket Weaving Building at 12:00 noon today, where the temperature is a steady 65 degrees Fahrenheit. When found, the core temperature was 91.1 degrees Fahrenheit. Two hours later, at 2:00 PM, the core temperature had fallen to 86.9. Assuming that the body temperature was 98.6 at the time of death, use Newton's law of cooling to find the time of death. ROUND TO 2 DECIMAL PLACES. The time of death was about ______ hours before the body was found.

Learn how to apply Newton's Law of Cooling to estimate the time of death. This detailed solution explores core concepts in exponential decay and real-life applications of physics and calculus, using body temperature data to solve the problem.

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