dT Newton's law of cooling states that the rate of change in the temperature T(t) of a body is proportional to the difference between the temperature of the medium M(t) and the temperature of the body. That is, K[M(t)- T(t)], where K is a constant. Let K=0.04 (min) and the temperature of the medium be constant, M(t) = 291 kelvins. If the body is initially at 365 kelvins, use Euler's method with h = 0.1 min to approximate the temperature of the body after (a) 30 minutes and (b) 60 minutes. (a) The temperature of the body after 30 minutes is kelvins. (Round to two decimal places as needed.) C
dT Newton's law of cooling states that the rate of change in the temperature T(t) of a body is proportional to the difference between the temperature of the medium M(t) and the temperature of the body. That is, K[M(t)- T(t)], where K is a constant. Let K=0.04 (min) and the temperature of the medium be constant, M(t) = 291 kelvins. If the body is initially at 365 kelvins, use Euler's method with h = 0.1 min to approximate the temperature of the body after (a) 30 minutes and (b) 60 minutes. (a) The temperature of the body after 30 minutes is kelvins. (Round to two decimal places as needed.) C
Advanced Engineering Mathematics
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Author:Erwin Kreyszig
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![### Understanding Newton's Law of Cooling
Newton's law of cooling states that the rate of change in the temperature \( T(t) \) of a body is proportional to the difference between the temperature of the medium \( M(t) \) and the temperature of the body. Mathematically, this can be represented as:
\[ \frac{dT}{dt} = K(M(t) - T(t)) \]
where \( K \) is a constant that signifies the cooling rate. In this specific case, let \( K = 0.04 \) (min\(^{-1}\)) and the temperature of the medium be constant at \( M(t) = 291 \) kelvins.
If the body is initially at \( 365 \) kelvins, we will use Euler's method with \( h = 0.1 \) min to approximate the temperature of the body after:
(a) 30 minutes and
(b) 60 minutes.
For the given values, we will follow the iterative approach of Euler's method to find the temperature at given time intervals.
**(a) The temperature of the body after 30 minutes is \(\_\_\_\_\_\_\_\_\_\) kelvins. (Round to two decimal places as needed.)**
Here is a step-by-step explanation of the process if we were to conduct the calculations using Euler's method:
1. Start with \( T(0) = 365 \) kelvins.
2. Apply the formula using Euler's method repeatedly with the time step \( h = 0.1 \) min for each iteration.
3. Repeat the process until you reach the desired time (total of 300 iterations for 30 minutes).
**(b) The temperature of the body after 60 minutes is \(\_\_\_\_\_\_\_\_\_\) kelvins. (Round to two decimal places as needed.)**
The same process as explained above can be repeated for 60 minutes, resulting in a total of 600 iterations.
This approach provides an approximation for the temperature at the specified times.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc45acf51-cefc-42df-83e4-535a253da58c%2Ff27a28be-9413-42ec-ad31-5274f9990b81%2F2nt4xex9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Newton's Law of Cooling
Newton's law of cooling states that the rate of change in the temperature \( T(t) \) of a body is proportional to the difference between the temperature of the medium \( M(t) \) and the temperature of the body. Mathematically, this can be represented as:
\[ \frac{dT}{dt} = K(M(t) - T(t)) \]
where \( K \) is a constant that signifies the cooling rate. In this specific case, let \( K = 0.04 \) (min\(^{-1}\)) and the temperature of the medium be constant at \( M(t) = 291 \) kelvins.
If the body is initially at \( 365 \) kelvins, we will use Euler's method with \( h = 0.1 \) min to approximate the temperature of the body after:
(a) 30 minutes and
(b) 60 minutes.
For the given values, we will follow the iterative approach of Euler's method to find the temperature at given time intervals.
**(a) The temperature of the body after 30 minutes is \(\_\_\_\_\_\_\_\_\_\) kelvins. (Round to two decimal places as needed.)**
Here is a step-by-step explanation of the process if we were to conduct the calculations using Euler's method:
1. Start with \( T(0) = 365 \) kelvins.
2. Apply the formula using Euler's method repeatedly with the time step \( h = 0.1 \) min for each iteration.
3. Repeat the process until you reach the desired time (total of 300 iterations for 30 minutes).
**(b) The temperature of the body after 60 minutes is \(\_\_\_\_\_\_\_\_\_\) kelvins. (Round to two decimal places as needed.)**
The same process as explained above can be repeated for 60 minutes, resulting in a total of 600 iterations.
This approach provides an approximation for the temperature at the specified times.
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