A roasted turkey is taken from an oven when its temperature has reached 185 Fahrenheit and is placed on a table in a room where the temperature is 75 Fahrenheit. Give answers accurate to at least 2 decimal places.

(a) If the temperature of the turkey is 142 Fahrenheit after half an hour, what is its temperature after 45 minutes?

To solve this problem, we'll use **Newton's Law of Cooling**, which states that the rate of change of the temperature of an object is proportional to the difference between its temperature and the ambient temperature. Mathematically, it's expressed as:

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

where:
- \( T(t) \) is the temperature of the object at time \( t \).
- \( T_{\text{ambient}} \) is the ambient temperature.
- \( k \) is a constant.

The solution to this differential equation is:

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

where:
- \( T_0 \) is the initial temperature of the object at \( t = 0 \).
- \( t \) is the time elapsed.

Given:
- Initial temperature \( T_0 = 185^\circ \text{F} \)
- Ambient temperature \( T_{\text{ambient}} = 75^\circ \text{F} \)
- Temperature after 30 minutes \( T(30) = 142^\circ \text{F} \)

First, we need to find the constant \( k \) using the given information. Then, we can use that value to find the temperature after 45 minutes.

### Step 1: Find \( k \)
Using the formula for \( T(t) \):

\[
142 = 75 + (185 - 75)e^{-30k}
\]

Simplify and solve for \( k \):

\[
142 - 75 = 110e^{-30k}
\]

\[
67 = 110e^{-30k}
\]

\[
\frac{67}{110} = e^{-30k}
\]

\[
\ln\left(\frac{67}{110}\right) = -30k
\]

\[
k = -\frac{1}{30} \ln\left(\frac{67}{110}\right)
\]

### Step 2: Find the temperature after 45 minutes
Now, using the value of \( k \) we calculated, we substitute it into the formula to find \( T(45) \):

\[
T(45) = 75 + 110e^{-45k}
\]

Let's calculate this step by step.

### Calculation
I'll compute the value of \( k \) and then find \( T(45) \).import math

# Given values
T_ambient = 75  # Ambient temperature
T0 = 185  # Initial temperature of the turkey
T_30 = 142  # Temperature after 30 minutes
t1 = 30  # Time in minutes (half an hour)
t2 = 45  # Time in minutes (45 minutes)

# Calculate the constant k
k = -(1 / t1) * math.log((T_30 - T_ambient) / (T0 - T_ambient))

# Calculate the temperature after 45 minutes
T_45 = T_ambient + (T0 - T_ambient) * math.exp(-k * t2)
T_45The temperature of the turkey after 45 minutes is approximately **127.29°F**.

Would you like more details or have any other questions?

Here are some related questions you might find interesting:
1. How would the temperature change after 1 hour?
2. What if the ambient temperature was different?
3. How does the rate constant \( k \) affect the cooling process?
4. What happens if the turkey was removed from the oven at a different initial temperature?
5. How could this model be adjusted for different cooling conditions?

**Tip:** Newton's Law of Cooling can be applied to various scenarios, such as cooling beverages or even forensic investigations to estimate the time of death.

Learn how to solve a temperature change problem using Newton's Law of Cooling with a step-by-step solution. Given initial and ambient temperatures, calculate the temperature of a turkey after 45 minutes based on its cooling rate. Suitable for high school students studying differential equations.

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