To solve the problem of finding an appropriate sinusoidal function that models Omaha's monthly temperature variation and using it to estimate the normal temperature during the month of April, we will explore various given sinusoidal models. Each model will then be evaluated at [tex]\( t = 4 \)[/tex] (corresponding to the month of April). Here are the detailed steps and results:
### Model 1:
[tex]\[ a. \quad y = 28 \cos \left(\frac{\pi}{6} t - \frac{7 \pi}{6}\right) + 49 \][/tex]
#### Evaluating for [tex]\( t = 4 \)[/tex]:
[tex]\[ b. \quad y(4) = 28 \cos \left(\frac{\pi}{6} \cdot 4 - \frac{7 \pi}{6}\right) + 49 \][/tex]
[tex]\[ y(4) = 49 \][/tex]
So, the normal temperature during April using model 1 is [tex]\( 49^\circ \)[/tex].
### Model 2:
[tex]\[ a. \quad y = 49 \cos \left(\frac{\pi}{6} t - \frac{7 \pi}{6}\right) + 28 \][/tex]
#### Evaluating for [tex]\( t = 4 \)[/tex]:
[tex]\[ b. \quad y(4) = 49 \cos \left(\frac{\pi}{6} \cdot 4 - \frac{7 \pi}{6}\right) + 28 \][/tex]
[tex]\[ y(4) = 28 \][/tex]
So, the normal temperature during April using model 2 is [tex]\( 28^\circ \)[/tex].
### Model 3:
[tex]\[ a. \quad y = 28 \sin \left(\frac{\pi}{6} t - \frac{7 \pi}{6}\right) + 49 \][/tex]
#### Evaluating for [tex]\( t = 4 \)[/tex]:
[tex]\[ b. \quad y(4) = 28 \sin \left(\frac{\pi}{6} \cdot 4 - \frac{7 \pi}{6}\right) + 49 \][/tex]
[tex]\[ y(4) = 21 \][/tex]
So, the normal temperature during April using model 3 is [tex]\( 21^\circ \)[/tex].
### Model 4:
[tex]\[ a. \quad y = 28 \cos \left(\frac{\pi}{12} t + \frac{7 \pi}{12}\right) + 49 \][/tex]
#### Evaluating for [tex]\( t = 4 \)[/tex]:
[tex]\[ b. \quad y(4) = 28 \cos \left(\frac{\pi}{12} \cdot 4 + \frac{7 \pi}{12}\right) + 49 \][/tex]
[tex]\[ y(4) \approx 21.95 \][/tex]
So, the normal temperature during April using model 4 is approximately [tex]\( 21.95^\circ \)[/tex].
### Summary:
- Model 1: [tex]\( y(4) = 49^\circ \)[/tex]
- Model 2: [tex]\( y(4) = 28^\circ \)[/tex]
- Model 3: [tex]\( y(4) = 21^\circ \)[/tex]
- Model 4: [tex]\( y(4) \approx 21.95^\circ \)[/tex]
Based on these models, the estimated normal temperatures during the month of April vary depending on the function chosen. The correct or best fit model would generally be determined by error analysis compared to actual historical data, but for the given answer, it seems that model 1 and model 3 provide temperatures close to 49 degrees as one prominent estimate.
