# Math Help - Harmonic Motion and Sinusoidal Function Problem....

1. ## Harmonic Motion and Sinusoidal Function Problem....

I have a huge math portfolio due tomorrow, and I could really use someone's help figuring out my last problem.

Thanks!

Suppose the length of time between two consecutive high tides is approximately 12.1 hours. According to NOAA on June 28, 1997 in Juneau, Alaska, high tide occurred at 8:11 AM (8.1833 hours) and low tide occurred at 2:14 PM (14.2333 hours). The height of the water at high tide was 13.2 feet and the height of the water at low tide was 2.2 feet.

a) Find a sinusoidal function in the form y=AsinB(x+C)+D for the situation.
b) Approximately when will the next high tide occur?

2. Hello sweetsyd101

Welcome to Math Help Forum!
Originally Posted by sweetsyd101
I have a huge math portfolio due tomorrow, and I could really use someone's help figuring out my last problem.

Thanks!

Suppose the length of time between two consecutive high tides is approximately 12.1 hours. According to NOAA on June 28, 1997 in Juneau, Alaska, high tide occurred at 8:11 AM (8.1833 hours) and low tide occurred at 2:14 PM (14.2333 hours). The height of the water at high tide was 13.2 feet and the height of the water at low tide was 2.2 feet.

a) Find a sinusoidal function in the form y=AsinB(x+C)+D for the situation.
b) Approximately when will the next high tide occur?
Assuming that we measure $t$ from 0:00 hours on June 28:

When $t = 8.1833, y$ is at its maximum value. Since the maximum value of the sine function is $+1$, this gives:
$13.2 = A + D$
When $t = 14.233, y$ is at its minimum value. So:
$2.2 = -A + D$
Solve these two equations for $A$ and $D$. (Easy!)

The time between consecutive high tides, $12.1$, is the period of the function. This is $\frac{2\pi}{B}$ for the function $A\sin B(x+C)$. So solve the equation
$\frac{2\pi}{B}=12.1$
to get the value of $B$.

Finally, the easiest way to get the value of $C$ is to work back $\frac{12.1}{4}$ hours (= time between high time and the time of the mid-point between the tides) to get the value of $x$ when the sine function has the value zero, and $y$ is increasing.

This give the value $x=8.133 - 3.025 = 5.108$ hours. So $C =-5.108$, since this will give the value $\sin B(x+C) = 0$, when $x = 5.108$.