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.
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
To model the variation in inches above and below the average water level as a function of time, we can use a cosine function. Let's define the midline as the average water level.
Given that the water level varies 70 inches between low tide at 8:40 AM and high tide at 2:55 PM, we have a total variation of 70 inches.
Now, let's find the period of our cosine function, which is the time it takes to complete one full cycle. The time between low tide and high tide is 6 hours and 15 minutes, or 375 minutes. Since one full cycle represents the time between two identical points, we can say the period is 375 minutes.
To convert this period to hours, we divide by 60: 375/60 = 6.25 hours.
The standard form of the cosine function is given as:
f(x) = A * cos(B * (x - C)) + D
In our case, the amplitude A is half the total variation, which is 70/2 = 35 inches.
The period B is simply 2π divided by the period in hours, so B = 2π/6.25.
The phase shift C is equal to 0 since we start measuring time from 8:40 AM.
The vertical shift D is equal to the midline, which is the average water level.
Therefore, the equation representing the variation in inches above and below the average water level is:
f(x) = 35 * cos((2π/6.25)(x - 0)) + D
To find when the function crosses the midline, we need to determine when f(x) = D. In this case, that would be when the water level is at the average level.
The midline represents the average water level. It is the horizontal line that divides the graph into two equal parts, with the water level above the midline corresponding to positive values and below the midline corresponding to negative values.
By solving the equation f(x) = D, you can determine the exact time when the function crosses the midline, which will indicate the point in the cycle.