Question

The depth of the water at the end of a pier changes periodically along with the movement of tides. On a particular day, low tides occur at 12:00 am and 12:30 pm, with a depth of 2.5 m, while high tides occur at 6:15 am and 6:45 pm, with a depth of 5.5 m. Let t = 0 be 12:00 am. Which periodic function, sine or cosine, would be a simpler model for the situation? Explain.

Answer 1

Refer to the figure shown below which illustrates the problem.
Low tides occur at a depth of 2.5 m at 12:00 am and 12:30 pm, separated by a period of T = 12.5 hours.
Likewise, high tides occur at a depth of 5.5 m at 6:15 am and 6:45 pm, separated by a period of T = 12.5 hours.

Use a time coordinate of t hours.
The amplitude is (1/2)*(5.5 - 2.5) = 1.5 m.
Use x(t) to denote depth at time t.

Because x(0)=2.5 and x(T/2) = 5.5,, use the periodic function
$x(t)=-1.5\,cos( \frac{2 \pi t}{12.5} )+4$

Verify the model.
x(0) = -1.5 + 4 = 2.5
x(6.25) = 1.5 + 4 = 5.5
x(12.5) = -1.5 + 4 = 2.5

A computer plot for x(t) is shown in the accompanying figure.

Answer 2

Answer: A cosine function would be a simpler model for the situation.

The minimum depth (low tide) occurs at

t = 0. A reflection of the cosine curve also has a minimum at t = 0.

A sine model would require a phase shift, while a cosine model does not.

Step-by-step explanation: