We need to find four parameters (numbers), namely A, B, C, and D. And we need to choose + or - sine or cosine function.
A, the amplitude, controls the total height of the sinusoid's graph. Notice the high point is 14' above the base and the low point is 2 ' above. This gives a total height of 12' (the difference). Divide that difference by 2 to get A = 6. Those two numbers also give us D, the middle line (or the axis, or the average value) by taking their average, which is 8' (halfway between 2 and 14).
Notice those two numbers are about vertical heights, y-values, so they can be considered the vertical stretch and shift, if you are thinking in terms of function transformations.
Next, we can get B by using the formula B = 2π / period. It is 6.5 hrs between high and low tides, so 13 hrs for a full cycle (peak to peak). B = 2π/13. Lastly, the most convenient choice of sinusoid is the + cosine function, which starts at its peak value. If, as the question suggests, we use midnight as t = 0, the peak occurs at t = 2, which means the cosine function is shifted right by 2 units, and C = 2. Again, in terms of transformations, these two values, B and C, represent a horizontal shrink and a horizontal stretch.
h(t) = 6cos(2π/13(t - 2)) + 8
To find the water's height at a specified time, simply evaluate the function by plugging in that # for time, as in b
If instead, as in part c, you are asked for specific times when the water will be a specified height, this will require plugging in that given height for h and solving by using inverse trig.
For example:
6 = 6cos(2π/13(t - 2)) + 8
-2 = 6cos(2π/13(t - 2))
-1/3 = cos(2π/13(t - 2))
cos-1(-1/3) = 2π/13(t - 2). etc.
Be sure your calculator is in radians mode. You will get the 1st time the water reaches 6' (on its way toward low tide). To find the time when the water rises again to 6', use the symmetry of the graph: subtract our 1st answer from 8.5 (time of low tide), then add that difference to 8.5 to find the time when the people need to stop drawing.
