MATH

posted by .

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!

  • MATH -

    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

  • Alert MATH -

    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

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. Physics

    Initially, you hear the siren at 90 dB. When the distance between you and the ambulance is a quarter of what it was initially, at what intensity level do you hear the siren?
  2. physical

    a police siren emits a sinusoidal wave with frequency f=300hz.the speed of sound is 340ms. find the wave lenght of the waves if the siren is at rest in the air
  3. 8th grade science

    The loudness of sound can be measured in decibels. Do you think loudness is a vector or not?
  4. physics

    Residents of Hawaii are warned of the approach of a tsunami by sirens mounted on the tops of towers. Suppose a siren produces a sound that has an intensity level of 120 dB at a distance of 2.0 m. How far away can the siren be heard?
  5. physical

    You are driving along a highway at 27.7 m/s when you hear a siren. You look in the rear-view mirror and see a police car approaching you from behind with a constant speed. The frequency of the siren that you hear is 1830. Hz. Right …
  6. physics

    at a distance of 3.8 m from a siren, the sound intensity is 3.6 x 10-2 W/m2. Assuming that the siren radiates sound uniformly in all directions, find the total power radiated.
  7. physics

    You're in an auto traveling at 25.0m/s toward a pole mounted warning siren. If the siren's frequency is 365 Hz, what frequency do you hear?
  8. math

    The ratio of the loudness of Lana’s earphones to the loudness of Polina’s earphones to the loudness of Goffy’s earphones is 6:5:3. If Polina’s earphones have a loudness of 420 decibels, find the total loudness of all three …
  9. physics

    A marine siren consists of a cylinder 72 cm in length closed at one end.for sound through air at 336 m/s. find the lowest frequency of the siren
  10. Physics

    A siren emits a sound of frequency 1200 kHz. With which frequency is it received by a car moving at a speed of 20 m/s. a) Away from the siren?

More Similar Questions