Question

suppose that the water level varies 70 inches between low tide at 8:.40 AM and high tide at 2:55PM .what he cosine function that models the variation in inches above and below the average water level as a function of the number of hours since 8:40AM .at what point in the cycle does the function cross the midline.what does the midline represent.

Answer 1

recall that the most general equation of a cosine curve is
y = a cos k(x - h) + d

I assume you know what each of these parameters represent

"water level varies 70 inches between low tide at 8:.40 AM and high tide at 2:55PM"
----> a = 35

low = 8:40 am, high = 2:55 pm, so 1/2 of a period = 14:55 - 8:40 = 6:15
so my period is 12:30 or 12.5 hrs

period = 2π/k
k = 2π/12.5 = 4π/25

so far we would have y = 35 cos (4π/25)(t ) + d
I think I will let d = 0, so the midline would be the midpoint between high and low tide.

The standard cosine has a max of 1, when t = 0 , and drops as we move to the right.
we want our equation to have a value of 35 when t = 8:40 or t = 26/3
so we have to move our normal cosine 26/3 to the right

---> y = 35 cos (4π/25)(t - 26/3)

let's test it:
when t = 26/3 , y = 35cos(4π/25)(0) = 35(1) = 35 , check!
when t = 14:55 = 179/12 , y = 35cos(4π/25)(25/4) = 35(-1) = -35 , check!
how about half way:
t = 283/24
y = 35cos(4π/25)(25/8) = 35cos(1.57079...) = 35(0) = 0 , yeahhh!!

My equation is correct.

further proof:
http://www.wolframalpha.com/input/?i=plot+y+%3D+35+cos+((4%CF%80%2F25)(x+-+26%2F3)),+for+0%3Cx%3C24