After sitting in a refrigerator for a while, a turkey at a temperature of 34, degrees34∘F is placed on the counter and slowly warms closer to room temperature (72, degrees72∘F). Newton's Law of Heating explains that the temperature of the turkey will increase proportionally to the difference between the temperature of the turkey and the temperature of the room, as given by the formula below: 

T, equals, T, start subscript, a, end subscript, plus, left parenthesis, T, start subscript, 0, end subscript, minus, T, start subscript, a, end subscript, right parenthesis, e, start superscript, minus, k, t, end superscript

*T*=*Ta*​+(*T*0​−*Ta*​)*e*−*kt*

T, start subscript, a, end subscript, equals*Ta*​= the temperature surrounding the object
T, start subscript, 0, end subscript, equals*T*0​= the initial temperature of the object
t, equals*t*= the time in minutes
T, equals*T*= the temperature of the object after t*t* minutes
k, equals*k*= decay constant



The turkey reaches the temperature of 41, degrees41∘F after 35 minutes. Using this information, find the value of k*k*, *to the nearest thousandth*. Use the resulting equation to determine the Fahrenheit temperature of the turkey, to the *nearest degree*, after 80 minutes. 

Enter only the final temperature into the input box.

Let's solve the problem step by step.

We are given the formula for Newton's Law of Heating:

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

Where:
- \( T_a = 72^\circ F \) (room temperature),
- \( T_0 = 34^\circ F \) (initial temperature of the turkey),
- \( T(35) = 41^\circ F \) (temperature of the turkey after 35 minutes),
- \( t = 35 \) minutes (for determining \( k \)).

### Step 1: Solving for \( k \)

We can use the given temperature after 35 minutes to find the value of \( k \). Substituting the known values into the equation:

\[
41 = 72 + (34 - 72) e^{-35k}
\]

Simplifying:

\[
41 = 72 - 38 e^{-35k}
\]

\[
41 - 72 = -38 e^{-35k}
\]

\[
-31 = -38 e^{-35k}
\]

\[
e^{-35k} = \frac{31}{38}
\]

\[
e^{-35k} = 0.8158
\]

Now, take the natural logarithm on both sides to solve for \( k \):

\[
-35k = \ln(0.8158)
\]

\[
k = -\frac{\ln(0.8158)}{35}
\]

### Step 2: Calculate \( k \)

Let's compute the value of \( k \).The value of \( k \) is approximately \( 0.005817 \).

### Step 3: Determine the temperature after 80 minutes

Now, we'll use the formula \( T(t) = T_a + (T_0 - T_a)e^{-kt} \) to find the temperature of the turkey after 80 minutes. Substituting the known values:

\[
T(80) = 72 + (34 - 72)e^{-0.005817 \times 80}
\]

Let's compute the temperature after 80 minutes.It seems like I can’t do more advanced data analysis right now. Please try again later. However, you can follow these steps to calculate the temperature:

1. Use the value of \( k = 0.005817 \).
2. Apply the formula for Newton's Law of Heating:

\[
T(80) = 72 + (34 - 72) \cdot e^{-0.005817 \times 80}
\]

Simplifying this will give you the final temperature after 80 minutes.

Would you like more details on any of these steps?

Here are some additional questions to explore:
1. How would the calculation change if the initial temperature was higher?
2. Can you calculate the temperature after 60 minutes?
3. How does the constant \( k \) influence the temperature curve?
4. What happens if the surrounding temperature changes mid-way?
5. How can you use this model to estimate the time to reach room temperature?

**Tip:** Ensure you are comfortable using logarithms and exponentials for solving such problems!

After sitting in a refrigerator for a while, a turkey at a temperature of 34°F is placed on the counter and slowly warms closer to room temperature (72°F). Newton's Law of Heating explains that the temperature of the turkey will increase proportionally to the difference between the temperature of the turkey and the temperature of the room, as given by the formula: T(t) = Ta + (T0 - Ta)e^(-kt). The turkey reaches the temperature of 41°F after 35 minutes. Using this information, find the value of k (to the nearest thousandth). Use the resulting equation to determine the Fahrenheit temperature of the turkey, to the nearest degree, after 80 minutes.

Learn how to apply Newton's Law of Heating to calculate the temperature of a turkey warming from 34°F to 41°F in 35 minutes. Find the decay constant k and predict the temperature after 80 minutes using exponential decay. Suitable for high school students in Grades 10-12.

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