Ta =the temperature surrounding the object To = the initial temperature of the object t = the time in minutes T = the temperature of the object after t minutes | k = decay constant The turkey reaches the temperature of 49°F after 25 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 115 minutes. Enter only the final temperature into the input box.
Ta =the temperature surrounding the object To = the initial temperature of the object t = the time in minutes T = the temperature of the object after t minutes | k = decay constant The turkey reaches the temperature of 49°F after 25 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 115 minutes. Enter only the final temperature into the input box.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![**Understanding Newton's Law of Heating**
When a turkey at a temperature of 40°F is removed from a refrigerator and placed on the counter, it warms up towards the room temperature of 73°F. According to Newton's Law of Heating, the increase in the temperature of the turkey is proportional to the difference between its initial temperature and the room temperature. The formula representing this relationship is:
\[ T = T_a + (T_0 - T_a)e^{-kt} \]
Where:
- \( T_a \) = the temperature of the surroundings (room temperature)
- \( T_0 \) = the initial temperature of the object (turkey)
- \( t \) = time in minutes
- \( T \) = the temperature of the object at time \( t \)
This exponential formula shows how the temperature of the turkey gradually increases over time, approaching but never quite reaching the room temperature.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F607cfd73-341e-40b9-9da6-af1be2ce5002%2F2ff815a4-8f14-441e-a4cc-36c15656ec49%2F7klq9f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Understanding Newton's Law of Heating**
When a turkey at a temperature of 40°F is removed from a refrigerator and placed on the counter, it warms up towards the room temperature of 73°F. According to Newton's Law of Heating, the increase in the temperature of the turkey is proportional to the difference between its initial temperature and the room temperature. The formula representing this relationship is:
\[ T = T_a + (T_0 - T_a)e^{-kt} \]
Where:
- \( T_a \) = the temperature of the surroundings (room temperature)
- \( T_0 \) = the initial temperature of the object (turkey)
- \( t \) = time in minutes
- \( T \) = the temperature of the object at time \( t \)
This exponential formula shows how the temperature of the turkey gradually increases over time, approaching but never quite reaching the room temperature.
![**Transcription and Explanation for Educational Use**
### Variables and Definitions:
- \( T_a \) = the temperature surrounding the object
- \( T_0 \) = the initial temperature of the object
- \( t \) = the time in minutes
- \( T \) = the temperature of the object after \( t \) minutes
- \( k \) = decay constant
### Problem Statement:
The turkey reaches the temperature of 49°F after 25 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 115 minutes.
Enter only the final temperature into the input box.
### Explanation:
This problem likely refers to Newton's Law of Cooling, which describes the cooling of a warmer object to the cooler temperature of the surrounding environment. The formula is:
\[ T(t) = T_a + (T_0 - T_a) \times e^{-kt} \]
Using the information provided (temperature at 25 minutes), calculate the decay constant \( k \), and then determine the temperature at 115 minutes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F607cfd73-341e-40b9-9da6-af1be2ce5002%2F2ff815a4-8f14-441e-a4cc-36c15656ec49%2Flg3gb3e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Transcription and Explanation for Educational Use**
### Variables and Definitions:
- \( T_a \) = the temperature surrounding the object
- \( T_0 \) = the initial temperature of the object
- \( t \) = the time in minutes
- \( T \) = the temperature of the object after \( t \) minutes
- \( k \) = decay constant
### Problem Statement:
The turkey reaches the temperature of 49°F after 25 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 115 minutes.
Enter only the final temperature into the input box.
### Explanation:
This problem likely refers to Newton's Law of Cooling, which describes the cooling of a warmer object to the cooler temperature of the surrounding environment. The formula is:
\[ T(t) = T_a + (T_0 - T_a) \times e^{-kt} \]
Using the information provided (temperature at 25 minutes), calculate the decay constant \( k \), and then determine the temperature at 115 minutes.
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