Given: y = A cos(Bx + C) + D. Throughout the day the depth of water at the end of a pier varies with the tides. High tide occurs at 5:00 a.m. with a depth of 5 feet. Low tide occurs at 10:00 a.m. with a depth of -.5 feet.
At what times will there be at least 2.25 feet of water at the dock, assuming 0 represents the depth at low tide.
You have not shown any work, but these can be tough the first time through, so here goes.
Always start by plotting points for the given information. Your y-axis is water height, and your x-axis is time, t.
When t = 5, y = 5; plot the point. This is a high point.
When t = 10, y = -.5; plot the point. This is a low point.
Can you sketch in the sinusoidal curve between these two points? What you have sketched is one half of a sinusoidal waveform, in other words, half a period/wave.
The amplitude is half the difference between the high and low points, so
A = (5 – (-5))/2 = 2.75
Find the “middle” of your wave by either adding the amplitude to the low point or subtracting the amplitude from the high point:
D = 5 – 2.75 = 2.25
The period must contain one complete wave, but so far we have only sketched one half wave between t = 5 and t = 10. Therefore, our period twice this amount of time, so our period is
P = 2(10 – 5) = 10 hours
The relationship between “B” in your equation and P is that P = (2pi)/B or B = (2pi)/P, so
B = (2pi)/10 = pi/5
The final variable , D, has to do with the phase shift (a horizontal translation of the function – a slide sideways). You have written your general form as
y = A cos(Bx + C) + D. This could be rewritten as y = A cos[B(x + C/B)] + D. The phase shift is C/B.
Now here is an interesting point to remember: a sinusoidal function can be written as either a sine function or a cosine function. Since you have written “cos” in your equation, we will use that.
The simple “parent” function is y = cosx and starts with a high point on the y-axis. Our first high point at (5,5) has been slid 5 units to the right. That means our phase shift is 5, so C/B = 5.
Thus, our function is
y = A cos[B(x + C/B)] + D
y = 2.75 cos[(pi/5)(x - 5)] + 2.25
Finally, always check you work by plugging in the two points you started with and making sure they give the correct answers.
Hopefully, you can finish from here.
