A person who was listening to a siren reported that the frequency of the sound fluctuated with time, measured in seconds. The minimum frequency that the person heard was 400 Hz, and the maximum frequency was 1100 Hz. The maximum frequency occurred at t=0 and t=12. The person also reported that, in 12 seconds, she heard the maximum frequency 4 times (including the times at t=0 and t=12). What is the equation of the cosine function that describes the frequency of this siren?
To find the equation of the cosine function that describes the frequency of the siren, we can start by determining the period (T) and the amplitude (A) of the cosine function.
The period (T) of a cosine function is the time it takes for the function to complete one full cycle. In this case, the maximum frequency occurs at t=0 and t=12, so the time it takes for the siren to go from the maximum frequency to the maximum frequency again is 12 seconds. Therefore, the period (T) is 12 seconds.
The amplitude (A) of a cosine function is the maximum value reached by the function. In this case, the maximum frequency reported by the person is 1100 Hz. Therefore, the amplitude (A) is 1100 Hz.
The general equation for a cosine function is given by: y = A * cos(2π/T * t + ϕ), where y is the function value, t is the time, A is the amplitude, T is the period, and ϕ is the phase shift.
Since the maximum frequency occurs at t=0, we know that the phase shift (ϕ) is 0. Therefore, the equation for the cosine function that describes the frequency of the siren is: y = 1100 * cos(2π/12 * t).
Please note that the equation assumes that the time (t) is measured in seconds and the frequency (y) is measured in Hz.
To find the equation of the cosine function that describes the frequency of the siren, we need to examine the given information.
We know that the minimum frequency heard was 400 Hz, and the maximum frequency was 1100 Hz. This gives us the amplitude of the cosine function, which is half the difference between the maximum and minimum frequencies.
Amplitude (A) = (1100 Hz - 400 Hz) / 2 = 350 Hz
We are also given that the maximum frequency occurred at t=0 and t=12, with 4 occurrences in total within 12 seconds. This tells us the period of the cosine function, which is the time it takes to complete one full cycle. In this case, the period is 12 seconds.
Period (T) = 12 seconds
Finally, we can determine the phase shift (c) of the cosine function by looking at when the maximum frequency occurs initially (at t=0). Since the maximum frequency occurs at t=0, we have:
Phase shift (c) = 0
Putting all of this information together, we can construct the equation of the cosine function:
f(t) = A * cos(2π/T * (t - c))
Substituting the values we found earlier:
f(t) = 350 Hz * cos(2π/12 * t)
Simplifying further:
f(t) = 350 Hz * cos(π/6 * t)