Let's address each part of the question step-by-step.
### Part (a): Write a Sinusoidal Function
To write a sinusoidal function that models Omaha's monthly temperature variation, we will use the general form of a sinusoidal function, which can be written as:
[tex]\[ T(t) = A \sin\left(\frac{2\pi}{P} (t - \phi)\right) + M \][/tex]
where:
- [tex]\( T(t) \)[/tex] is the temperature at month [tex]\( t \)[/tex].
- [tex]\( A \)[/tex] is the amplitude, which represents half the range of temperature variations.
- [tex]\( P \)[/tex] is the period, which for a yearly cycle is 12 months.
- [tex]\( \phi \)[/tex] is the phase shift, which shifts the curve horizontally to align with the temperature data.
- [tex]\( M \)[/tex] is the midline, which is the average temperature over the year.
#### Step 1: Calculate the Midline ([tex]\( M \)[/tex])
The midline [tex]\( M \)[/tex] can be calculated as the average of the maximum and minimum temperatures.
Given the temperatures: [21, 27, 39, 52, 62, 72, 77, 74, 65, 53, 39, 25],
- Maximum temperature = [tex]\( 77^{\circ}F \)[/tex]
- Minimum temperature = [tex]\( 21^{\circ}F \)[/tex]
The midline [tex]\( M \)[/tex] is given by:
[tex]\[ M = \frac{\text{Max Temp} + \text{Min Temp}}{2} = \frac{77 + 21}{2} = 49^{\circ}F \][/tex]
#### Step 2: Calculate the Amplitude ([tex]\( A \)[/tex])
The amplitude [tex]\( A \)[/tex] is half the difference between the maximum and minimum temperatures.
[tex]\[ A = \frac{\text{Max Temp} - \text{Min Temp}}{2} = \frac{77 - 21}{2} = 28^{\circ}F \][/tex]
#### Step 3: Determine the Period ([tex]\( P \)[/tex])
The period [tex]\( P \)[/tex] for our scenario is the length of one full cycle and, since the data is for a yearly cycle, [tex]\( P = 12 \)[/tex] months.
#### Step 4: Calculate the Phase Shift ([tex]\( \phi \)[/tex])
To find the phase shift, we locate the month with the peak temperature.
The highest temperature [tex]\( 77^{\circ}F \)[/tex] occurs in July (month 7). For a standard sine function, the peak would occur at [tex]\( t = \frac{P}{4} \)[/tex] (3 months).
Therefore, the phase shift [tex]\( \phi \)[/tex] aligns this with month 7:
[tex]\[ \phi = 7 - 3 = 4 \][/tex]
#### Final Sinusoidal Function
With all these values, the sinusoidal function that models Omaha’s monthly temperature variation is:
[tex]\[ T(t) = 28 \sin\left(\frac{2\pi}{12} (t - 4)\right) + 49 \][/tex]
### Part (b): Estimate the Normal Temperature in April
To estimate the normal temperature for the month of April using the sinusoidal model, we substitute [tex]\( t = 4 \)[/tex] into our function.
[tex]\[ T(4) = 28 \sin\left(\frac{2\pi}{12} (4 - 4)\right) + 49 \][/tex]
Simplifying inside the sine function:
[tex]\[ T(4) = 28 \sin(0) + 49 \][/tex]
Since [tex]\( \sin(0) = 0 \)[/tex]:
[tex]\[ T(4) = 28 \cdot 0 + 49 = 49 \][/tex]
Therefore, the estimated normal temperature during the month of April is [tex]\( 49^{\circ}F \)[/tex].
