Question 1150036
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t = number of hours that has elapsed since some starting point
H(t) = height, in meters, of water above the low tide level
H(t) = 8 means we're 8 meters above the low tide level
H(t) = 2 means we're 2 meters above the low tide level
The smallest H(t) can get is 0. The largest is 8.
The range of this trig function is {{{0 <= H(t) <= 8}}} which is the same as saying {{{0 <= y <= 8}}} if y = H(t)


General sine function is
y = A*sin(B(x-C))+D
A = amplitude
B = determined by the period
C = phase shift (aka horizontal shift)
D = vertical shift = midline


Lets find the values for A,B,C,D
A = amplitude
A = half the vertical distance between the lower and upper bounds of the range
A = (upper - lower)/2
A = (8-0)/2
A = 4


P = period = 36.5 hours
B = 2pi/P
B = 2pi/36.5


C = phase shift
We'll come back to this later


D = midline
D = midpoint of upper and lower boundaries of the range
D = (upper+lower)/2
D = (8+0)/2
D = 4


We have
A = 4
B = 2pi/36.5
C = unknown for now
D = 4


So,
{{{y = A*sin(B(x-C))+D}}}
turns into
{{{y = 4*sin(expr(2pi/36.5)(x-C))+4}}}


Let's plug in (x,y) = (0,2) to indicate we want the starting water level to be 2 meters.
In other words, we want H(t) = 2 when t = 0.
Now solve for the variable C
{{{y = 4*sin(expr(2pi/36.5)(x-C))+4}}}
{{{2 = 4*sin(expr(2pi/36.5)(0-C))+4}}}
{{{2-4 = 4*sin(expr(2pi/36.5)(-C))}}}
{{{-2 = 4*sin(-expr(2pi/36.5)C)}}}
{{{4*sin(-expr(2pi/36.5)C) = -2}}}
{{{sin(-expr(2pi/36.5)C) = -2/4}}}
{{{sin(-expr(2pi/36.5)C) = -1/2}}}
{{{-expr(2pi/36.5)C = arcsin(-1/2)}}}
{{{-expr(2pi/36.5)C = -pi/6}}}
{{{expr(2/36.5)C = 1/6}}}
{{{C = (36.5/2)*(1/6)}}}
{{{C = 18.25*(1/6)}}}
{{{C = 18.25/6}}}

We can update 
{{{y = 4*sin(expr(2pi/36.5)(x-C))+4}}}
into
{{{y = 4*sin(expr(2pi/36.5)(x-18.25/6))+4}}}
{{{y = 4*sin(expr(2pi/36.5)x-expr(2pi/36.5)*(18.25/6))+4}}}
{{{y = 4*sin(expr(2pi/36.5)x-pi/6)+4}}}
{{{H(t) = 4*sin(expr(2pi/36.5)t-pi/6)+4}}}


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Final Answer:
{{{H(t) = 4*sin(expr(2pi/36.5)t-pi/6)+4}}}
Graph:
<img src = "https://i.imgur.com/5oRCAHE.png">
Notes:
<ul>
<li>x = t = elapsed time in hours</li>
<li>y = H(t) = height, in meters, of water above the low tide level</li>
<li>The sine curve stays between y = 0 and y = 8</li>
<li>The graph repeats itself every x = 36.5 units. The water is at a height of y = 2 when x = 0. The water level goes up, comes back down to y = 0, and goes back up again to arrive back at y = 2 when x = 36.5</li>
<li>The horizontal distance between any peak to its adjacent neighbor valley is x = 18.25, which is half of the period 36.5</li>
<li>The amplitude of 4 represents vertical distance from the midline y = 4 to either the peak (y = 8) or the valley (y = 0).</li>
</ul>
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