The average monthly temperature changes from month to month. Suppose that, for a given city, we can model the average temperature in a month with the following function.
f(t)= 59 - 26 sin (pi/6 t)
In this equation, f(t) is the average temperature in a month in degrees Fahrenheit, and t is the month of the year (January=1, February=2, ...).
Find the following. If necessary, round to the nearest hundredth.
Number of cycles of f per month :
period T is when (pi/6) t = 2 pi
t = 12 months = period T = 1 year period
f = frequency = 1/T = 1/12 cycle/month (logically enough)
I do not believe it is the mean in Jan and June though. You need a phase angle in there.
To find the number of cycles of the function f(t) per month, we need to determine the period of the sinusoidal function.
For any sinusoidal function of the form f(t) = A * sin(B * (t - C)) + D, the period can be found using the formula:
Period (P) = 2π / |B|
In our given function f(t) = 59 - 26 * sin(π/6 * t), we can see that the coefficient of t is B = π/6.
Now, let's substitute this value into the formula to find the period (P):
P = 2π / |π/6|
P = 2π * 6/π
P = 12
So, the period of the function f(t) is 12.
Since we are interested in the number of cycles per month, we can conclude that there are 12 cycles of f(t) per month.
To find the number of cycles of the function f(t) per month, we need to determine the period of the function.
The general form of a sine function is f(t) = A sin(Bt + C) + D, where:
A represents the amplitude,
B represents the period,
C represents the phase shift, and
D represents the vertical shift.
In the given equation, f(t) = 59 - 26 sin(pi/6 * t), the coefficient in front of t in the sine function is (pi/6), which represents the period of the function.
The formula to find the period (T) of a sine function is T = 2pi/B.
Substituting the value of B = (pi/6) into the formula, we get:
T = 2pi / (pi/6)
T = 12
Therefore, the period of the function f(t) is 12 months.
Since there are 12 months in a year, each month represents one cycle of the function f(t).
Hence, the number of cycles of f per month is 1.