Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Newton’s Law of Cooling: Understanding Temperature Changes**
After sitting on a shelf for a while, a can of soda at room temperature (73°F) is placed inside a refrigerator and slowly cools. The temperature of the refrigerator is 35°F. Newton's Law of Cooling explains that the temperature of the can of soda will decrease proportionally to the difference between the temperature of the can and the temperature of the refrigerator, 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
**Explanation of the Formula:**
The formula above describes how the temperature \( T \) of an object approaches the ambient temperature \( T_a \). The expression \( (T_0 - T_a)e^{-kt} \) represents the exponential decay of the temperature difference over time. The rate at which the temperature changes is determined by the constant \( k \), which depends on the properties of the object and the conditions of cooling.](https://content.bartleby.com/qna-images/question/607cfd73-341e-40b9-9da6-af1be2ce5002/179ba41b-0336-4669-9d09-74d0f8f3d744/2tkl7la_processed.jpeg)
Transcribed Image Text:**Newton’s Law of Cooling: Understanding Temperature Changes**
After sitting on a shelf for a while, a can of soda at room temperature (73°F) is placed inside a refrigerator and slowly cools. The temperature of the refrigerator is 35°F. Newton's Law of Cooling explains that the temperature of the can of soda will decrease proportionally to the difference between the temperature of the can and the temperature of the refrigerator, 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
**Explanation of the Formula:**
The formula above describes how the temperature \( T \) of an object approaches the ambient temperature \( T_a \). The expression \( (T_0 - T_a)e^{-kt} \) represents the exponential decay of the temperature difference over time. The rate at which the temperature changes is determined by the constant \( k \), which depends on the properties of the object and the conditions of cooling.
![**Newton's Law of Cooling**
\[ T = T_a + (T_0 - T_a) e^{-kt} \]
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 can of soda reaches the temperature of 53°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 can of soda, to the nearest degree, after 70 minutes.](https://content.bartleby.com/qna-images/question/607cfd73-341e-40b9-9da6-af1be2ce5002/179ba41b-0336-4669-9d09-74d0f8f3d744/trtbgbd_processed.jpeg)
Transcribed Image Text:**Newton's Law of Cooling**
\[ T = T_a + (T_0 - T_a) e^{-kt} \]
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 can of soda reaches the temperature of 53°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 can of soda, to the nearest degree, after 70 minutes.
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