Danny is measuring the loudness of a police siren. At t=2 seconds, the siren's loudness is at its maximum of 112 dB for the first time. At t=7 seconds, the siren's loudness is at its minimum of 88 dB for the first time. The loudness is a sinusoidal function of time, t. In the first 13 seconds, how much of the time will the loudness be above 94 dB?
So I know the sinusoidal function is this:
y=Asin(2pi/B(x-C)+D
A=amplitude=12
D=Mean=100
B=period=???
C=phase shift= x-coordinate of max-(B/4)=?????
need help please!
your a is correct at 12
your period:
from max to min took 5 seconds, so the whole period is 10 seconds
then 10 = 2pi/k
k = pi/5
so the basic curve could be
y = 12sin(pi/5)(x)
this would have a max of 12 but you want the max to be 112 , so let's move it up 100 units
y = 12sin(pi/5)(x) + 100
Right now our max would be at 2.5 seconds, but we want it to be at 2 seconds, so our graph must be shifted to the left .5 units.
Final curve:
y = 12sin(pi/5)(x + .5) + 100
testing:
if t=2, y = 112 , check!
if t=7, y = 88 check!
now let's set y = 94
94 = 12sin(pi/5)(x + .5) + 100
-.5 = sin(pi/5)(x + .5)
(pi/5)(x + .5) = pi+.526599 = 3.66519
x = 5.33333
or
(pi/5)(x + .5) = 2pi-.526599 = 5.75959
x = 8.6667
The next set of answers would be 10 seconds later, (the period is 10), but you wanted it only in the first 13 seconds
So between the times of 5.33 and 8.67 seconds we are below 94 decibels, (looking at our graph)
so for a time period of 8.67-5.33 or 3.34 seconds we are below 94 decibels, which means that in the first 13 seconds, the sound is above
94 decibels for a time of 13-3.34 seconds or 9.66 seconds
another check:
take a time between the above, say t = 5.5
y = 12sin(pi/5)(5.5 + .5) + 100
= 92.9 which is below 94
got my above and belows mixed up in one line
6th last line should read:
means that in the first 13 seconds, the sound is below
To determine the period, B, of the sinusoidal function in this case, we can use the given information.
The loudness reaches its maximum of 112 dB at t=2 seconds. Since the function is periodic, the loudness will repeat after a certain interval. Therefore, we need to find the difference in time between two consecutive maximum points.
From the given information, we know that at t=2 seconds, the loudness is at its maximum for the first time. Let's assume this is one complete cycle of the sinusoidal function. Therefore, the next maximum point will occur after one full cycle, which is the period B.
Now, we need to find the difference in time between two consecutive maximum points. Since the second maximum point occurs at t=2+B (after one full cycle), the difference in time will be (t=2+B) - (t=2) = B.
Therefore, the period, B, is equal to the time difference between two consecutive maximum points. In this case, B = (t=2+B) - (t=2), which simplifies to B = B.
To find the phase shift, C, we need to determine the x-coordinate of the maximum point relative to the starting point of the period. In this case, the maximum point occurs at t=2 seconds. So, the phase shift, C, can be calculated as:
C = x-coordinate of max - (B/4) = 2 - (B/4)
Now, let's substitute the known values A=12, D=100, and the equations for period B and phase shift C, into the sinusoidal function formula:
y = 12sin(2pi/B(x-C)) + 100
To determine how much of the time in the first 13 seconds the loudness will be above 94 dB, we need to solve the inequality:
12sin(2pi/B(x-C)) + 100 > 94
To solve this inequality, you can plot the graph of this function and find where it is above 94 dB on the interval between t=0 and t=13 seconds. Alternatively, you can evaluate the function for different values of t within this interval and see when the loudness is above 94 dB.
Remember to consider the period B and phase shift C calculated earlier when evaluating the function for different values of t.