A thermometer is removed from a room where the temperature is 70° F and is taken outside, where the air temperature is 19° F. After one-half minute the thermometer reads 52° F. What is the reading of the thermometer at t = 1 min? Assume that the rate of change of temperature in time is proportional to the difference between the thermometer's temperature and the air temperature. Type your answer in °F in the space provided below. Round your answer to two decimal places.
A thermometer is removed from a room where the temperature is 70° F and is taken outside, where the air temperature is 19° F. After one-half minute the thermometer reads 52° F. What is the reading of the thermometer at t = 1 min? Assume that the rate of change of temperature in time is proportional to the difference between the thermometer's temperature and the air temperature. Type your answer in °F in the space provided below. Round your answer to two decimal places.
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![### Thermometer Temperature Change Problem
**Problem Statement:**
A thermometer is removed from a room where the temperature is 70°F and is taken outside, where the air temperature is 19°F. After one-half minute, the thermometer reads 52°F. What is the reading of the thermometer at \( t = 1 \) minute? Assume that the rate of change of temperature in time is proportional to the difference between the thermometer’s temperature and the air temperature.
_Type your answer in °F in the space provided below. Round your answer to two decimal places._
[Answer Box Here]
**Explanation:**
To solve this problem, we can use Newton's Law of Cooling, which states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the ambient temperature (in this case, the outside air temperature).
The formula is given by:
\[ \frac{dT}{dt} = k(T - T_{\text{ambient}}) \]
Where:
- \( T \) is the temperature of the object (the thermometer).
- \( T_{\text{ambient}} \) is the ambient temperature (outside air temperature, 19°F).
- \( k \) is the proportionality constant.
- \( t \) is the time.
Given initial conditions:
- Initial temperature of the thermometer (\( T(0) \)) = 70°F
- Temperature after 0.5 minutes (\( T(0.5) \)) = 52°F
We need to find the temperature at \( t = 1 \) minute.
1. We use these initial conditions to find \( k \).
2. Solve for \( T \) at \( t = 1 \) minute.
This will involve solving a differential equation and applying the initial conditions to find the particular solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2e521a1e-f4ee-4c02-af71-8dfc2cc184d3%2F9d33386b-daa2-494c-aebc-e1653f029c21%2F8dsfh3f_processed.png&w=3840&q=75)
Transcribed Image Text:### Thermometer Temperature Change Problem
**Problem Statement:**
A thermometer is removed from a room where the temperature is 70°F and is taken outside, where the air temperature is 19°F. After one-half minute, the thermometer reads 52°F. What is the reading of the thermometer at \( t = 1 \) minute? Assume that the rate of change of temperature in time is proportional to the difference between the thermometer’s temperature and the air temperature.
_Type your answer in °F in the space provided below. Round your answer to two decimal places._
[Answer Box Here]
**Explanation:**
To solve this problem, we can use Newton's Law of Cooling, which states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the ambient temperature (in this case, the outside air temperature).
The formula is given by:
\[ \frac{dT}{dt} = k(T - T_{\text{ambient}}) \]
Where:
- \( T \) is the temperature of the object (the thermometer).
- \( T_{\text{ambient}} \) is the ambient temperature (outside air temperature, 19°F).
- \( k \) is the proportionality constant.
- \( t \) is the time.
Given initial conditions:
- Initial temperature of the thermometer (\( T(0) \)) = 70°F
- Temperature after 0.5 minutes (\( T(0.5) \)) = 52°F
We need to find the temperature at \( t = 1 \) minute.
1. We use these initial conditions to find \( k \).
2. Solve for \( T \) at \( t = 1 \) minute.
This will involve solving a differential equation and applying the initial conditions to find the particular solution.
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