Final answer:
To model Omaha's monthly temperature using a sinusoidal function, we can use a trigonometric equation in the form of y = A sin(B(x - C)) + D, where A represents the amplitude, B represents the frequency, C represents the phase shift, and D represents the vertical shift.
Step-by-step explanation:
To model Omaha's monthly temperature using a sinusoidal function, we can use a trigonometric equation in the form of y = A sin(B(x - C)) + D, where A represents the amplitude, B represents the frequency, C represents the phase shift, and D represents the vertical shift.
In this case, the temperature range for Omaha is between 61°F and 95°F. We can determine the amplitude by calculating half the difference between the maximum and minimum temperatures, which is (95 - 61) / 2 = 17°F. The frequency can be determined by dividing 2π by the number of days in the month, which is roughly 30. The phase shift can be determined by finding the day value that corresponds to the minimum temperature, subtracting 1 to account for the day index starting at 0, and then dividing by the number of days in the month (30). Lastly, the vertical shift can be determined by finding the average of the maximum and minimum temperatures, which is (95 + 61) / 2 = 78°F.
Putting it all together, the sinusoidal function that models Omaha's monthly temperature is:
y = 17 sin((2π/30)(x - k)) + 78, where k represents the day of the month.