Let H = f(t, v) refer to the temperature of the machine (oC), which %3| depends on the elapsed time the machine is running t (hours) and the production speed (cookies per minute). Some values of f(t, v) are given in the table below. tv || 20 | 30 40 50 60 1 20 21 24 30 35 2 26 27 31 38 45 28 29 35 44 | 47 4 29 30 37 45 48 45 | 48 30 37 30 | 37 5 29 6. 29 45 | 48 (a) Use a central difference to estimate the value fp(4, 40). (b) Interpret the value of f,(4, 40) above in context of the given situation.
Let H = f(t, v) refer to the temperature of the machine (oC), which %3| depends on the elapsed time the machine is running t (hours) and the production speed (cookies per minute). Some values of f(t, v) are given in the table below. tv || 20 | 30 40 50 60 1 20 21 24 30 35 2 26 27 31 38 45 28 29 35 44 | 47 4 29 30 37 45 48 45 | 48 30 37 30 | 37 5 29 6. 29 45 | 48 (a) Use a central difference to estimate the value fp(4, 40). (b) Interpret the value of f,(4, 40) above in context of the given situation.
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Please help with answering both parts (a) and (b).
![## Temperature of the Machine Based on Elapsed Time and Production Speed
Let \( H = f(t, v) \) refer to the temperature of the machine (°C), which depends on the elapsed time the machine is running \( t \) (hours) and the production speed \( v \) (cookies per minute). Some values of \( f(t, v) \) are given in the table below.
| \( t \backslash v \) | 20 | 30 | 40 | 50 | 60 |
|---------------------|----|----|----|----|----|
| 1 | 20 | 21 | 24 | 30 | 35 |
| 2 | 26 | 27 | 31 | 38 | 45 |
| 3 | 28 | 29 | 35 | 44 | 47 |
| 4 | 29 | 30 | 37 | 45 | 48 |
| 5 | 29 | 30 | 37 | 45 | 48 |
| 6 | 29 | 30 | 37 | 45 | 48 |
(a) **Use a central difference to estimate the value \( f_v(4, 40) \).**
To estimate \( f_v(4, 40) \) using the central difference, we use the values at \( v = 30 \) and \( v = 50 \) for \( t = 4 \):
\[ f_v(4, 40) \approx \frac{f(4, 50) - f(4, 30)}{50 - 30} = \frac{45 - 30}{20} = \frac{15}{20} = 0.75 \]
Therefore, \( f_v(4, 40) \approx 0.75 \).
(b) **Interpret the value of \( f_v(4, 40) \) above in the context of the given situation.**
The value \( f_v(4, 40) = 0.75 \) represents the rate of change of the machine's temperature with respect to the production speed at \( t = 4 \) hours and \( v = 40 \) cookies per minute. Specifically, it means that increasing the production speed by one cookie per minute at this point in time increases the temperature by approximately](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0194bc48-d116-4697-91f6-6559bfb11fe2%2F87c12aa5-4fda-451e-a25a-22a76b251dd4%2Fat6ks2_processed.png&w=3840&q=75)
Transcribed Image Text:## Temperature of the Machine Based on Elapsed Time and Production Speed
Let \( H = f(t, v) \) refer to the temperature of the machine (°C), which depends on the elapsed time the machine is running \( t \) (hours) and the production speed \( v \) (cookies per minute). Some values of \( f(t, v) \) are given in the table below.
| \( t \backslash v \) | 20 | 30 | 40 | 50 | 60 |
|---------------------|----|----|----|----|----|
| 1 | 20 | 21 | 24 | 30 | 35 |
| 2 | 26 | 27 | 31 | 38 | 45 |
| 3 | 28 | 29 | 35 | 44 | 47 |
| 4 | 29 | 30 | 37 | 45 | 48 |
| 5 | 29 | 30 | 37 | 45 | 48 |
| 6 | 29 | 30 | 37 | 45 | 48 |
(a) **Use a central difference to estimate the value \( f_v(4, 40) \).**
To estimate \( f_v(4, 40) \) using the central difference, we use the values at \( v = 30 \) and \( v = 50 \) for \( t = 4 \):
\[ f_v(4, 40) \approx \frac{f(4, 50) - f(4, 30)}{50 - 30} = \frac{45 - 30}{20} = \frac{15}{20} = 0.75 \]
Therefore, \( f_v(4, 40) \approx 0.75 \).
(b) **Interpret the value of \( f_v(4, 40) \) above in the context of the given situation.**
The value \( f_v(4, 40) = 0.75 \) represents the rate of change of the machine's temperature with respect to the production speed at \( t = 4 \) hours and \( v = 40 \) cookies per minute. Specifically, it means that increasing the production speed by one cookie per minute at this point in time increases the temperature by approximately
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