anonymous
  • anonymous
26. The water depth in a harbor is 25 m at high tide occurring at midnight, and 9 m at low tide. One cycle is completed approximately every 12 hours. a) Find a sine and cosine equation for the water depth as a function of time, t hours starting at high tide
Mathematics
katieb
  • katieb
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anonymous
  • anonymous
i just need help with creating the sine equation I got some values already I got A = 25 , B = 30, D = 17 i just wanna find the value C ] y = Asin[B(x - C)] + D
anonymous
  • anonymous
The amplitude of the wave A, is the maximum displacement from the average height:\[A=\frac{25-9}{2}\] the frequency, given by 2pi divided by the period of the wave should be \[B = \frac{2\pi}{12}\] C can equal 0 in this case since you're starting at high tide if you assume a cosine function. and the value for D should be the displacement from the origin. or the difference between your amplitude and you're actual highest point. \[D=25-8=17\]
anonymous
  • anonymous
so how do i find C

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anonymous
  • anonymous
for sine function
anonymous
  • anonymous
use the identity sin(x+pi/2)=cos(x)
anonymous
  • anonymous
can you do that for me
anonymous
  • anonymous
.... if you have \[A\sin (B x+C) + D = A \cos( B x )+D\] then C must be pi/2
anonymous
  • anonymous
y = 8sin[30(x - C)] + 17 tell me how would i solve for C and can you show me your answer
anonymous
  • anonymous
C = 9 according to my answer sheet
anonymous
  • anonymous
ah jeeze, I'm sorry... this is what I get for not understanding the question fully, C = 270 degrees in degrees, but you want how many hours its offset by in order for it to be a sine wave instead of a cosine wave right? so what you'd want to do it displace the wave off by what would be 270 degrees, but since a full rotation is 360 degrees, which is how you got that 30 there, you would take 360 and multiply it by 3/4 giving you 270 which when you factor that 30 out you get the 9 that your answer sheet gives Sorry I didn't get it at first.
anonymous
  • anonymous
how do you know the wave has to be ofsetted by 270 degrees and not any other value
anonymous
  • anonymous
that's that identity i was talking about, when you switch Pi/2 from radians into degrees it's equal to 90 degrees, which when you subract from 360 gives you 270.

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