in many parts of the coutry the average temp of a particular area may be modeled by a sin usoidal function. According to NCDC the average temp of k city is 92.2 in july and 32.2 in january assume that the period is 12 what is the best model for when july is month 0 and when january is month 0
To find the best model for the average temperature fluctuations in a particular area, which is modeled by a sinusoidal function, we need to determine the equation of the function.
The standard form of a sinusoidal function is:
f(x) = A * sin(B(x - C)) + D
Where:
A represents the amplitude (half the difference between the maximum and minimum values)
B represents the frequency (in this case, it will be 2π divided by the period)
C represents the horizontal shift (the phase shift or the month at which the temperature starts its cycle)
D represents the vertical shift (the average temperature that the sinusoidal function is centered around)
In this case, we are given the average temperature of a particular city in July (month 0) as 92.2 and in January (month 6) as 32.2. Therefore, there is a horizontal shift of 6 months.
Now, let's calculate the amplitude and frequency:
Amplitude (A):
The amplitude can be found by taking half the difference between the maximum and minimum temperatures provided:
Amplitude (A) = (92.2 - 32.2) / 2 = 60 / 2 = 30
Frequency (B):
The frequency can be calculated using the formula: 2π / Period
Given that the period is 12 months, we have:
Frequency (B) = 2π / 12 = π / 6
Now we can determine the equation of the sinusoidal function by substituting the values we obtained into the standard form equation:
f(x) = 30 * sin((π / 6)(x - 6)) + C
To find the vertical shift (C), we need to determine the average temperature that the sinusoidal function is centered around. In this case, the average temperature is given as 62.2, which is the average of the maximum and minimum temperatures provided:
Vertical Shift (C) = (92.2 + 32.2) / 2 = 124.4 / 2 = 62.2
Now we can write the final equation for the temperature model when July is month 0 and January is month 6:
f(x) = 30 * sin((π / 6)(x - 6)) + 62.2
This equation represents the best model for the average temperature fluctuations in the given city.