After sitting in a refrigerator for a while, a turkey at a temperature of 36°F is placed on the counter and slowly warms closer to room temperature (69°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 = Ta + (To – Ta)e-k -kt || 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 44°F after 20 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 60 minutes. Enter only the final temperature into the input box.
After sitting in a refrigerator for a while, a turkey at a temperature of 36°F is placed on the counter and slowly warms closer to room temperature (69°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 = Ta + (To – Ta)e-k -kt || 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 44°F after 20 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 60 minutes. Enter only the final temperature into the input box.
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![**Understanding Newton's Law of Heating**
After sitting in a refrigerator for a while, a turkey at a temperature of 36°F is placed on the counter and slowly warms closer to room temperature (69°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:
\[ T = T_a + (T_0 - T_a)e^{-kt} \]
where:
- \( T_a \) = 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
The turkey reaches the temperature of 44°F after 20 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 60 minutes.
**Step-by-Step Solution**
1. **Identify given values:**
- Initial temperature, \( T_0 = 36°F \)
- Surrounding temperature, \( T_a = 69°F \)
- Temperature after 20 minutes, \( T = 44°F \)
2. **Formulate the equation with given values:**
\[ 44 = 69 + (36 - 69)e^{-20k} \]
3. **Solve for \( k \):**
- Simplify the equation:
\[ 44 = 69 - 33e^{-20k} \]
\[ -25 = -33e^{-20k} \]
\[ \frac{25}{33} = e^{-20k} \]
- Take the natural logarithm (ln) of both sides to solve for \( k \):
\[ \ln\left(\frac{25}{33}\right) = -20k \]
\[ k = -\frac{\ln\left(\frac{25}{33}\right)}{20} \]
4. **Calculate \( k \):**
\[ k \approx 0.016 \text{ (to the nearest thousandth)} \]
5. **Use \( k \) to find the temperature after 60 minutes](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F488b60ff-9f36-4743-8e7c-aaf54d5d0a5e%2F3921ac76-ce0d-4521-9d72-d9f93f6f5ccd%2F8sks51_processed.png&w=3840&q=75)
Transcribed Image Text:**Understanding Newton's Law of Heating**
After sitting in a refrigerator for a while, a turkey at a temperature of 36°F is placed on the counter and slowly warms closer to room temperature (69°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:
\[ T = T_a + (T_0 - T_a)e^{-kt} \]
where:
- \( T_a \) = 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
The turkey reaches the temperature of 44°F after 20 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 60 minutes.
**Step-by-Step Solution**
1. **Identify given values:**
- Initial temperature, \( T_0 = 36°F \)
- Surrounding temperature, \( T_a = 69°F \)
- Temperature after 20 minutes, \( T = 44°F \)
2. **Formulate the equation with given values:**
\[ 44 = 69 + (36 - 69)e^{-20k} \]
3. **Solve for \( k \):**
- Simplify the equation:
\[ 44 = 69 - 33e^{-20k} \]
\[ -25 = -33e^{-20k} \]
\[ \frac{25}{33} = e^{-20k} \]
- Take the natural logarithm (ln) of both sides to solve for \( k \):
\[ \ln\left(\frac{25}{33}\right) = -20k \]
\[ k = -\frac{\ln\left(\frac{25}{33}\right)}{20} \]
4. **Calculate \( k \):**
\[ k \approx 0.016 \text{ (to the nearest thousandth)} \]
5. **Use \( k \) to find the temperature after 60 minutes
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