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 92.8 degrees Fahrenheit.  Two hours​ later, at​ 2:00 PM, the core temperature had fallen to 87.8.   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 can use **Newton's Law of Cooling** to solve this problem. According to Newton's Law of Cooling, the rate of change of the temperature of an object is proportional to the difference between the object's temperature and the surrounding temperature.

The formula is given by:

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

Where:
- \( T(t) \) is the temperature of the object at time \( t \),
- \( T_s \) is the surrounding temperature (65°F in this case),
- \( T_0 \) is the initial temperature (98.6°F, the body temperature at the time of death),
- \( k \) is the cooling constant,
- \( t \) is the time in hours since death.

We are given two pieces of information:
1. At \( t = 0 \) (12:00 noon), the temperature of the body was \( T(0) = 92.8°F \),
2. At \( t = 2 \) (2:00 PM), the temperature of the body was \( T(2) = 87.8°F \).

We need to find \( k \) first, then use it to find the time of death.

### Step 1: Solve for \( k \)
We'll use the two temperature data points to solve for \( k \).

I'll calculate this first.It seems like I can’t do more advanced data analysis right now. Please try again later. Would you like to proceed with another question, or should I guide you through solving this manually?

Here are a few follow-up questions to consider:
1. What are the assumptions of Newton's Law of Cooling?
2. How does the cooling constant \( k \) affect the cooling process?
3. Why is it important to know the surrounding temperature in such problems?
4. Can Newton's Law of Cooling be applied to heating scenarios?
5. How would the time of death calculation change if the surrounding temperature varied?

**Tip:** When solving temperature-based problems, tracking units (Fahrenheit, Celsius, etc.) is essential for accuracy.

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 92.8 degrees Fahrenheit. Two hours later, at 2:00 PM, the core temperature had fallen to 87.8. 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 use Newton's Law of Cooling to estimate the time of death in a real-world scenario. This problem involves exponential decay, body temperature, and ambient temperature. Suitable for students in Grades 10-12.

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