Question

The ocean tides near Carter Beach follow a repeating pattern over time, and can be

modeled using a cosine function as the amount of time between each low tide and
high tide is constant. On a given day, low tide occurred at 8:30 a.m. and high tide
occurred at 3:00 p.m. At high tide, the water level was 12 inches above the average
local sea level; at low tide it was 12 inches below the average local sea level. Assume
that high tide and low tide are the maximum and minimum water levels each day,
respectively.

People who fish in Carter Beach know that a certain species of fish is most plentiful
when the water level is increasing. Explain whether you would recommend fishing
for this species at 7:30 p.m. or 10:30 p.m. using evidence from the given context
above.

Answer 1

Explanation:

D, the vertical shift, also called the sinusoidal axis, or the average value, can be calculated by averaging the y-value of the high point and the y-value of the low point: D = (6 + 2) / 2 = 4.

A, the amplitude, will be the difference of the high and low point y-values divided by 2: (6 - 2) / 2 = 2. We should also think of A as how far away the high and low points are from the average value, D.

Next, we calculate B, the parameter controlling the length of the period by using the formula: B = 2π / period. We are told the 1st high tide is at 4 am, while the 1st low tide is at 10 am. These are 6 hrs apart, which means the highs are 12 hrs apart. Thus, B = 2π / 12 = π/6.

Lastly, an unshifted cosine function will begin at a peak, when x = 0. This cosine function has a peak when time = 4, meaning it is shifted right by 4 from a normal cosine curve. So C = - 4.

Putting all together, we get the following:

x: time elapsed since midnight (in hrs)

y: depth of water (in ft)

y = 2cos(π/6(x - 4)) + 4.

If you wanted to give the answer in the form your teacher requested, distribute the B-value across (x - 4), though the form above is more commonly used.