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Sagot :
Let's go through each of the sinusoidal functions step-by-step and determine which one correctly models the monthly temperature and the estimate for the temperature in April.
### Sinusoidal Models:
We have three sinusoidal models presented:
1. [tex]\( y = 28 \cos\left(\frac{\pi}{6} t - \frac{7\pi}{6}\right) + 49 \)[/tex]
2. [tex]\( y = 49 \cos\left(\frac{\pi}{6} t - \frac{7\pi}{6}\right) + 28 \)[/tex]
3. [tex]\( y = 28 \sin\left(\frac{\pi}{6} t - \frac{7\pi}{6}\right) + 49 \)[/tex]
### Estimating the temperature for April ([tex]\( t = 4 \)[/tex]):
#### Model 1: [tex]\( y = 28 \cos\left(\frac{\pi}{6} t - \frac{7\pi}{6}\right) + 49 \)[/tex]
For [tex]\( t = 4 \)[/tex]:
[tex]\[ y(4) = 28 \cos\left(\frac{\pi}{6} \cdot 4 - \frac{7\pi}{6}\right) + 49 \][/tex]
[tex]\[ y(4) = 28 \cos\left(\frac{4\pi}{6} - \frac{7\pi}{6}\right) + 49 \][/tex]
[tex]\[ y(4) = 28 \cos\left(\frac{-3\pi}{6}\right) + 49 \][/tex]
[tex]\[ y(4) = 28 \cos\left(-\frac{\pi}{2}\right) + 49 \][/tex]
[tex]\[ y(4) = 28 \cdot 0 + 49 \][/tex]
[tex]\[ y(4) = 49 \][/tex]
So, the estimated temperature for April using Model 1 is [tex]\( 49^\circ F \)[/tex].
#### Model 2: [tex]\( y = 49 \cos\left(\frac{\pi}{6} t - \frac{7\pi}{6}\right) + 28 \)[/tex]
For [tex]\( t = 4 \)[/tex]:
[tex]\[ y(4) = 49 \cos\left(\frac{\pi}{6} \cdot 4 - \frac{7\pi}{6}\right) + 28 \][/tex]
[tex]\[ y(4) = 49 \cos\left(\frac{4\pi}{6} - \frac{7\pi}{6}\right) + 28 \][/tex]
[tex]\[ y(4) = 49 \cos\left(\frac{-3\pi}{6}\right) + 28 \][/tex]
[tex]\[ y(4) = 49 \cos\left(-\frac{\pi}{2}\right) + 28 \][/tex]
[tex]\[ y(4) = 49 \cdot 0 + 28 \][/tex]
[tex]\[ y(4) = 28 \][/tex]
So, the estimated temperature for April using Model 2 is [tex]\( 28^\circ F \)[/tex].
#### Model 3: [tex]\( y = 28 \sin\left(\frac{\pi}{6} t - \frac{7\pi}{6}\right) + 49 \)[/tex]
For [tex]\( t = 4 \)[/tex]:
[tex]\[ y(4) = 28 \sin\left(\frac{\pi}{6} \cdot 4 - \frac{7\pi}{6}\right) + 49 \][/tex]
[tex]\[ y(4) = 28 \sin\left(\frac{4\pi}{6} - \frac{7\pi}{6}\right) + 49 \][/tex]
[tex]\[ y(4) = 28 \sin\left(\frac{-3\pi}{6}\right) + 49 \][/tex]
[tex]\[ y(4) = 28 \sin\left(-\frac{\pi}{2}\right) + 49 \][/tex]
[tex]\[ y(4) = 28 \cdot (-1) + 49 \][/tex]
[tex]\[ y(4) = -28 + 49 \][/tex]
[tex]\[ y(4) = 21 \][/tex]
So, the estimated temperature for April using Model 3 is [tex]\( 21^\circ F \)[/tex].
### Conclusion:
Comparing the results:
- Model 1: [tex]\( y(4) = 49^\circ F \)[/tex]
- Model 2: [tex]\( y(4) = 28^\circ F \)[/tex]
- Model 3: [tex]\( y(4) = 21^\circ F \)[/tex]
Given the results, we can see the following:
- For option 1: The function [tex]\( y = 28 \cos\left(\frac{\pi}{6} t - \frac{7\pi}{6}\right) + 49 \)[/tex] produces an estimate of [tex]\( 49^\circ F \)[/tex] for April.
- For option 3: The function [tex]\( y = 28 \sin\left(\frac{\pi}{6} t - \frac{7\pi}{6}\right) + 49 \)[/tex] produces an estimate of [tex]\( 21^\circ F \)[/tex] for April.
- Neither matches the given temperature [tex]\( 52^\circ \)[/tex] for April.
Given that [tex]\(49^\circ F \)[/tex] match with Model 1, the most reasonable conclusion is that Model 1 is a better representation of temperature variation.
### Final Answers:
a. The sinusoidal function that models Omaha's monthly temperature variation is [tex]\( y = 28 \cos\left(\frac{\pi}{6} t - \frac{7\pi}{6}\right) + 49 \)[/tex].
b. Using this model, the estimated normal temperature during the month of April is [tex]\( y(4) = 49^\circ F \)[/tex].
### Sinusoidal Models:
We have three sinusoidal models presented:
1. [tex]\( y = 28 \cos\left(\frac{\pi}{6} t - \frac{7\pi}{6}\right) + 49 \)[/tex]
2. [tex]\( y = 49 \cos\left(\frac{\pi}{6} t - \frac{7\pi}{6}\right) + 28 \)[/tex]
3. [tex]\( y = 28 \sin\left(\frac{\pi}{6} t - \frac{7\pi}{6}\right) + 49 \)[/tex]
### Estimating the temperature for April ([tex]\( t = 4 \)[/tex]):
#### Model 1: [tex]\( y = 28 \cos\left(\frac{\pi}{6} t - \frac{7\pi}{6}\right) + 49 \)[/tex]
For [tex]\( t = 4 \)[/tex]:
[tex]\[ y(4) = 28 \cos\left(\frac{\pi}{6} \cdot 4 - \frac{7\pi}{6}\right) + 49 \][/tex]
[tex]\[ y(4) = 28 \cos\left(\frac{4\pi}{6} - \frac{7\pi}{6}\right) + 49 \][/tex]
[tex]\[ y(4) = 28 \cos\left(\frac{-3\pi}{6}\right) + 49 \][/tex]
[tex]\[ y(4) = 28 \cos\left(-\frac{\pi}{2}\right) + 49 \][/tex]
[tex]\[ y(4) = 28 \cdot 0 + 49 \][/tex]
[tex]\[ y(4) = 49 \][/tex]
So, the estimated temperature for April using Model 1 is [tex]\( 49^\circ F \)[/tex].
#### Model 2: [tex]\( y = 49 \cos\left(\frac{\pi}{6} t - \frac{7\pi}{6}\right) + 28 \)[/tex]
For [tex]\( t = 4 \)[/tex]:
[tex]\[ y(4) = 49 \cos\left(\frac{\pi}{6} \cdot 4 - \frac{7\pi}{6}\right) + 28 \][/tex]
[tex]\[ y(4) = 49 \cos\left(\frac{4\pi}{6} - \frac{7\pi}{6}\right) + 28 \][/tex]
[tex]\[ y(4) = 49 \cos\left(\frac{-3\pi}{6}\right) + 28 \][/tex]
[tex]\[ y(4) = 49 \cos\left(-\frac{\pi}{2}\right) + 28 \][/tex]
[tex]\[ y(4) = 49 \cdot 0 + 28 \][/tex]
[tex]\[ y(4) = 28 \][/tex]
So, the estimated temperature for April using Model 2 is [tex]\( 28^\circ F \)[/tex].
#### Model 3: [tex]\( y = 28 \sin\left(\frac{\pi}{6} t - \frac{7\pi}{6}\right) + 49 \)[/tex]
For [tex]\( t = 4 \)[/tex]:
[tex]\[ y(4) = 28 \sin\left(\frac{\pi}{6} \cdot 4 - \frac{7\pi}{6}\right) + 49 \][/tex]
[tex]\[ y(4) = 28 \sin\left(\frac{4\pi}{6} - \frac{7\pi}{6}\right) + 49 \][/tex]
[tex]\[ y(4) = 28 \sin\left(\frac{-3\pi}{6}\right) + 49 \][/tex]
[tex]\[ y(4) = 28 \sin\left(-\frac{\pi}{2}\right) + 49 \][/tex]
[tex]\[ y(4) = 28 \cdot (-1) + 49 \][/tex]
[tex]\[ y(4) = -28 + 49 \][/tex]
[tex]\[ y(4) = 21 \][/tex]
So, the estimated temperature for April using Model 3 is [tex]\( 21^\circ F \)[/tex].
### Conclusion:
Comparing the results:
- Model 1: [tex]\( y(4) = 49^\circ F \)[/tex]
- Model 2: [tex]\( y(4) = 28^\circ F \)[/tex]
- Model 3: [tex]\( y(4) = 21^\circ F \)[/tex]
Given the results, we can see the following:
- For option 1: The function [tex]\( y = 28 \cos\left(\frac{\pi}{6} t - \frac{7\pi}{6}\right) + 49 \)[/tex] produces an estimate of [tex]\( 49^\circ F \)[/tex] for April.
- For option 3: The function [tex]\( y = 28 \sin\left(\frac{\pi}{6} t - \frac{7\pi}{6}\right) + 49 \)[/tex] produces an estimate of [tex]\( 21^\circ F \)[/tex] for April.
- Neither matches the given temperature [tex]\( 52^\circ \)[/tex] for April.
Given that [tex]\(49^\circ F \)[/tex] match with Model 1, the most reasonable conclusion is that Model 1 is a better representation of temperature variation.
### Final Answers:
a. The sinusoidal function that models Omaha's monthly temperature variation is [tex]\( y = 28 \cos\left(\frac{\pi}{6} t - \frac{7\pi}{6}\right) + 49 \)[/tex].
b. Using this model, the estimated normal temperature during the month of April is [tex]\( y(4) = 49^\circ F \)[/tex].
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