College Physics
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![**Heating Baby's Milk Using Newton's Law of Cooling**
Bob heats up a bottle of milk for his baby, Jill, every morning. Jill will only drink the bottle if it is at 98 degrees. So Bob places the bottle in a cup of hot water (170 degrees). The bottle starts off at 35 degrees. After 2 minutes, the bottle is 80 degrees.
Using Newton's law of cooling, which states that the rate of change in temperature, \( H \), is proportional to the difference between the object and the surrounding temperature, we can determine how long Bob should leave the bottle in the hot water. Assume that the water stays at 170 degrees.
Concepts Covered:
1. Newton's Law of Cooling
2. Temperature Change and Time Calculation
3. Practical Application in Daily Life
We’ll employ Newton's law of cooling formula:
\[ \frac{dT}{dt} = -k(T - T_{\text{env}}) \]
Where:
- \( T \) = temperature of the object at time \( t \)
- \( T_{\text{env}} \) = surrounding temperature (170 degrees in this case)
- \( k \) = constant of proportionality
- \( t \) = time
Steps:
1. Establish initial conditions.
- Initial temperature of the bottle, \( T_0 \) = 35 degrees
- Surrounding temperature, \( T_{\text{env}} \) = 170 degrees
- Temperature after 2 minutes, \( T(t=2) \) = 80 degrees
2. Use these conditions to solve for \( k \).
3. Calculate the time required for the bottle to reach 98 degrees using the derived constant \( k \).
By solving the differential equation with the provided initial conditions, we’ll ascertain the precise time Bob should leave the bottle in the hot water.
This system of equations and practical examples makes Newton's law of cooling comprehensible and applicable to everyday scenarios.](https://content.bartleby.com/qna-images/question/0b1d198d-95a1-4d0b-9915-fffff0f3c2fd/9a5963e6-b7a4-42e1-89f7-f5f343201b0c/6tye8mj_processed.png)
Transcribed Image Text:**Heating Baby's Milk Using Newton's Law of Cooling**
Bob heats up a bottle of milk for his baby, Jill, every morning. Jill will only drink the bottle if it is at 98 degrees. So Bob places the bottle in a cup of hot water (170 degrees). The bottle starts off at 35 degrees. After 2 minutes, the bottle is 80 degrees.
Using Newton's law of cooling, which states that the rate of change in temperature, \( H \), is proportional to the difference between the object and the surrounding temperature, we can determine how long Bob should leave the bottle in the hot water. Assume that the water stays at 170 degrees.
Concepts Covered:
1. Newton's Law of Cooling
2. Temperature Change and Time Calculation
3. Practical Application in Daily Life
We’ll employ Newton's law of cooling formula:
\[ \frac{dT}{dt} = -k(T - T_{\text{env}}) \]
Where:
- \( T \) = temperature of the object at time \( t \)
- \( T_{\text{env}} \) = surrounding temperature (170 degrees in this case)
- \( k \) = constant of proportionality
- \( t \) = time
Steps:
1. Establish initial conditions.
- Initial temperature of the bottle, \( T_0 \) = 35 degrees
- Surrounding temperature, \( T_{\text{env}} \) = 170 degrees
- Temperature after 2 minutes, \( T(t=2) \) = 80 degrees
2. Use these conditions to solve for \( k \).
3. Calculate the time required for the bottle to reach 98 degrees using the derived constant \( k \).
By solving the differential equation with the provided initial conditions, we’ll ascertain the precise time Bob should leave the bottle in the hot water.
This system of equations and practical examples makes Newton's law of cooling comprehensible and applicable to everyday scenarios.
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