A baked cake is taken out of a 375°F oven and placed on a table in a 72°F room. It cools according to Newton's Law. Ten minutes later the temperature is 345°F. d=k(T-A) T(t) = A +cekt a) Find the complete model T(t). b) What is the reading of the thermometer after 30 minutes? (Round your answer to the tenths place and use correct units.) c) How long does it take for the cake to cool to 100°F? (Round your answer to the tenths place and use correct units.)
A baked cake is taken out of a 375°F oven and placed on a table in a 72°F room. It cools according to Newton's Law. Ten minutes later the temperature is 345°F. d=k(T-A) T(t) = A +cekt a) Find the complete model T(t). b) What is the reading of the thermometer after 30 minutes? (Round your answer to the tenths place and use correct units.) c) How long does it take for the cake to cool to 100°F? (Round your answer to the tenths place and use correct units.)
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![**Newton's Law of Cooling: Application Problem**
A baked cake is taken out of a 375°F oven and placed on a table in a 72°F room. It cools according to Newton's Law. Ten minutes later the temperature is 345°F.
The differential equation describing the cooling process is given by:
\[ \frac{dT}{dt} = k(T - A) \]
The solution to the differential equation is:
\[ T(t) = A + ce^{kt} \]
### Problems:
**(a) Find the complete model \( T(t) \).**
**(b) What is the reading of the thermometer after 30 minutes?**
*(Round your answer to the tenths place and use correct units.)*
**(c) How long does it take for the cake to cool to 100°F?**
*(Round your answer to the tenths place and use correct units.)*
#### Explanation:
1. **Newton's Law of Cooling Equation:**
- \( \frac{dT}{dt} = k(T - A) \)
- \( T \) is the temperature of the object (cake).
- \( A \) is the ambient temperature.
- \( k \) is the cooling constant.
- The solution to this differential equation is \( T(t) = A + ce^{kt} \), where:
- \( c \) is a constant determined by initial conditions.
- \( t \) is time.
2. **Initial Conditions:**
- \( T(0) = 375^\circ\text{F} \) (initial temperature of the cake)
- \( A = 72^\circ\text{F} \) (ambient temperature)
- \( T(10) = 345^\circ\text{F} \) (temperature after 10 minutes)
3. **Finding the Complete Model \( T(t) \):**
- Use the initial conditions to find the constants \( c \) and \( k \).
- Substitute back into the model equation to get the complete expression for \( T(t) \).
4. **Thermometer Reading After 30 Minutes:**
- Substitute \( t = 30 \) into the complete model \( T(t) \) to find the temperature after 30 minutes.
5. **Time to Cool to 100°F:**
- Set \( T(t) =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F336090ec-d2ca-4d15-a2ae-85a71ff5580d%2Fc21cd0da-cd1a-45e4-96aa-1198c9b1d3bd%2Fg3lg3oi_processed.png&w=3840&q=75)
Transcribed Image Text:**Newton's Law of Cooling: Application Problem**
A baked cake is taken out of a 375°F oven and placed on a table in a 72°F room. It cools according to Newton's Law. Ten minutes later the temperature is 345°F.
The differential equation describing the cooling process is given by:
\[ \frac{dT}{dt} = k(T - A) \]
The solution to the differential equation is:
\[ T(t) = A + ce^{kt} \]
### Problems:
**(a) Find the complete model \( T(t) \).**
**(b) What is the reading of the thermometer after 30 minutes?**
*(Round your answer to the tenths place and use correct units.)*
**(c) How long does it take for the cake to cool to 100°F?**
*(Round your answer to the tenths place and use correct units.)*
#### Explanation:
1. **Newton's Law of Cooling Equation:**
- \( \frac{dT}{dt} = k(T - A) \)
- \( T \) is the temperature of the object (cake).
- \( A \) is the ambient temperature.
- \( k \) is the cooling constant.
- The solution to this differential equation is \( T(t) = A + ce^{kt} \), where:
- \( c \) is a constant determined by initial conditions.
- \( t \) is time.
2. **Initial Conditions:**
- \( T(0) = 375^\circ\text{F} \) (initial temperature of the cake)
- \( A = 72^\circ\text{F} \) (ambient temperature)
- \( T(10) = 345^\circ\text{F} \) (temperature after 10 minutes)
3. **Finding the Complete Model \( T(t) \):**
- Use the initial conditions to find the constants \( c \) and \( k \).
- Substitute back into the model equation to get the complete expression for \( T(t) \).
4. **Thermometer Reading After 30 Minutes:**
- Substitute \( t = 30 \) into the complete model \( T(t) \) to find the temperature after 30 minutes.
5. **Time to Cool to 100°F:**
- Set \( T(t) =
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