A turkey is taken out of the oven with an internal temperature of 165°F and is allowed to cool in a 75°F room. After half an hour, the internal temperature of the turkey is 145°F. To the nearest degree, what will the temperature be after 65

 minutes?

The temperature will be  °F
.

This is a classic example of **Newton's Law of Cooling**, which 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.

The formula for Newton's Law of Cooling is:

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

Where:
- \( T(t) \) is the temperature of the object at time \( t \),
- \( T_s \) is the surrounding temperature (in this case, 75°F),
- \( T_0 \) is the initial temperature of the object (165°F),
- \( k \) is the cooling constant,
- \( t \) is time (in hours).

### Step 1: Find the Cooling Constant \( k \)
We know that after half an hour (0.5 hours), the temperature is 145°F. So, we can plug the values into the formula to solve for \( k \).

\[
145 = 75 + (165 - 75) \cdot e^{-0.5k}
\]
\[
145 = 75 + 90 \cdot e^{-0.5k}
\]
\[
70 = 90 \cdot e^{-0.5k}
\]
\[
\frac{70}{90} = e^{-0.5k}
\]
\[
\frac{7}{9} = e^{-0.5k}
\]

Now, take the natural logarithm of both sides:

\[
\ln\left(\frac{7}{9}\right) = -0.5k
\]
\[
k = -2 \cdot \ln\left(\frac{7}{9}\right)
\]

### Step 2: Use \( k \) to Find the Temperature After 65 Minutes
Now that we have \( k \), we can plug it back into the formula to find the temperature after 65 minutes (which is \( t = \frac{65}{60} \) hours).

Let me calculate these steps for you.The temperature of the turkey after 65 minutes will be approximately **105°F**.

Would you like more details on any part of the calculation or have any other questions?

Here are 5 related questions for further exploration:
1. How would the cooling rate change if the room temperature were lower?
2. What if the initial temperature of the turkey was higher, say 180°F?
3. How does increasing the time affect the rate at which the turkey cools?
4. What would happen if the turkey were left for 90 minutes instead of 65 minutes?
5. Can we generalize this formula for any other cooling object in different ambient conditions?

**Tip**: When solving for exponential decay problems, keep in mind that the cooling rate depends heavily on the difference between the object and ambient temperatures.

A turkey is taken out of the oven with an internal temperature of 165°F and is allowed to cool in a 75°F room. After half an hour, the internal temperature of the turkey is 145°F. To the nearest degree, what will the temperature be after 65 minutes?

Solve a cooling problem using Newton's Law of Cooling, where a turkey cools from 165°F to 145°F in 30 minutes in a 75°F room. Find the temperature after 65 minutes with detailed explanations of exponential decay and calculus. Suitable for Grades 11-12.

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