A turkey is taken out of an oven and cools following Newton's Law of Cooling. The temperature of the room is held constant at 70 degrees. At 5:00 the temperature of the turkey is 140 degrees, and it cools to 110 degrees at 5:30. If the initial temperature of the turkey when it was removed from the oven was 180 degrees, at what time was the turkey taken from the oven?
A turkey is taken out of an oven and cools following Newton's Law of Cooling. The temperature of the room is held constant at 70 degrees. At 5:00 the temperature of the turkey is 140 degrees, and it cools to 110 degrees at 5:30. If the initial temperature of the turkey when it was removed from the oven was 180 degrees, at what time was the turkey taken from the oven?
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![**Educational Content on Newton's Law of Cooling**
**Example Problem: Cooling of a Turkey**
A turkey is taken out of an oven and cools following Newton's Law of Cooling. The temperature of the room is held constant at 70 degrees Fahrenheit. At 5:00 PM, the temperature of the turkey is 140 degrees, and it cools to 110 degrees by 5:30 PM. If the initial temperature of the turkey when it was removed from the oven was 180 degrees, at what time was the turkey taken from the oven?
**Explanation of Newton's Law of Cooling:**
Newton's Law of Cooling states that the rate at which an object cools is proportional to the difference in temperature between the object and its surrounding environment. The formula is generally expressed as:
\[ T(t) = T_{\text{env}} + (T_{\text{initial}} - T_{\text{env}}) e^{-kt} \]
Where:
- \( T(t) \) is the temperature of the object at time \( t \).
- \( T_{\text{env}} \) is the ambient temperature of the surrounding environment.
- \( T_{\text{initial}} \) is the initial temperature of the object.
- \( k \) is the cooling constant.
- \( t \) is the time elapsed.
**Steps to Solve the Problem:**
1. **Identify the given values:**
- \( T_{\text{env}} = 70^\circ \)
- \( T(0) = T_{\text{initial}} = 180^\circ \)
- \( T(1 \text{ hour}) = 140^\circ \)
- \( T(1.5 \text{ hours}) = 110^\circ \)
2. **Set up the equation using the given values:**
\[
140 = 70 + (180 - 70)e^{-kt}
\]
Simplify to solve for \( k \).
3. **Calculate the time when the turkey was taken from the oven using the temperature-time relationship and the calculated \( k \) value.**
This problem involves exponential decay and solving for time, incorporating principles of Newton's Law of Cooling. Understanding this law is essential in various fields such as physics, engineering, and culinary arts, helping to predict cooling times and temperature changes in practical scenarios.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcba7b9ea-03da-4ccc-8cbf-1d9caf29b38a%2F654df2ab-4cfc-409e-bb96-6e506265cea9%2Fu97r62_processed.png&w=3840&q=75)
Transcribed Image Text:**Educational Content on Newton's Law of Cooling**
**Example Problem: Cooling of a Turkey**
A turkey is taken out of an oven and cools following Newton's Law of Cooling. The temperature of the room is held constant at 70 degrees Fahrenheit. At 5:00 PM, the temperature of the turkey is 140 degrees, and it cools to 110 degrees by 5:30 PM. If the initial temperature of the turkey when it was removed from the oven was 180 degrees, at what time was the turkey taken from the oven?
**Explanation of Newton's Law of Cooling:**
Newton's Law of Cooling states that the rate at which an object cools is proportional to the difference in temperature between the object and its surrounding environment. The formula is generally expressed as:
\[ T(t) = T_{\text{env}} + (T_{\text{initial}} - T_{\text{env}}) e^{-kt} \]
Where:
- \( T(t) \) is the temperature of the object at time \( t \).
- \( T_{\text{env}} \) is the ambient temperature of the surrounding environment.
- \( T_{\text{initial}} \) is the initial temperature of the object.
- \( k \) is the cooling constant.
- \( t \) is the time elapsed.
**Steps to Solve the Problem:**
1. **Identify the given values:**
- \( T_{\text{env}} = 70^\circ \)
- \( T(0) = T_{\text{initial}} = 180^\circ \)
- \( T(1 \text{ hour}) = 140^\circ \)
- \( T(1.5 \text{ hours}) = 110^\circ \)
2. **Set up the equation using the given values:**
\[
140 = 70 + (180 - 70)e^{-kt}
\]
Simplify to solve for \( k \).
3. **Calculate the time when the turkey was taken from the oven using the temperature-time relationship and the calculated \( k \) value.**
This problem involves exponential decay and solving for time, incorporating principles of Newton's Law of Cooling. Understanding this law is essential in various fields such as physics, engineering, and culinary arts, helping to predict cooling times and temperature changes in practical scenarios.
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