# The depth of water in a harbour at high tide is 12m and at low tide is 4m. the water is at average depth at 2.00 am and high tide is..............coming in. Assuming a 12 hr cycle, determine sine...

The depth of water in a harbour at high tide is 12m and at low tide is 4m. the water is at average depth at 2.00 am and high tide is.........

.....coming in. Assuming a 12 hr cycle, determine sine and cosine functions ,that model the depth of water in the harbour as a function of time in hours after midnight?

lfryerda | Certified Educator

Considering a sine or cosine function that models the depth of water, the function will be in the form `y=a sin (Kt) + b` or `y = a cos (Kt) + b` , where `K={2 pi}/T` is a scaling coefficient, `T` is the period of the function, `a` is the amplitude, and `b` is the mean height.

In this case, high tide is 12m and low tide is 4m, so the amplitude is `a={12-4}/2=4m` , which means the average depth is `b=4+3=8m` .

Also, the period is `T=12` hours (given), and the average is at 2:00 am, so the depth is shifted to 2am for sine, and shifted to 8am for cosine.

Putting this altogether, we get the two (equivalent) functions `h=4sin({2 pi(t-2)}/12)+8` and `h=4cos({2 pi(t-8)}/12)+8` for the depth of the water in hours after midnight.