Tides go up and down in a 12.8 hour period. The average depth of the river is 9m, and ranges between 7 and 11 m. The variation can be approximated by a sine curve. Write an equation that gives the approximate y value, if x is the number of hours after midnight if high tide occurs at 6:00 am.
This is the equation for a sine curve; y = A sin (B(x-C)) +D
Y and X are the usual graph axes. We're letting Y describe the height of the wave (the amplitude) and X describes time. Therefore this sine curve should describe the height of the surface of the water over time.
A will represent the amplitude. The amplitude is given by the difference between the average height and the minimum and maximum heights. This value is 2.
B will represent the number of complete cycles that the wave goes through over a distance (in time) of 2pi. Since our horizontal axis is time, we need to find out how many 12.8 hour cycles can be completed in 2pi hours. 2pi/12.8 = .49087.
So far our equation looks like this: y = 2 sin (.49087(x-C)) +D
This is the equation for a wave with a period of 12.8 hours and a 2-meter amplitude. However, we need to adjust it so that it correctly reflects an average height of 9 meters, and a high point at the 6-hour mark. We will use C and D to adjust the graph along the x and y axes.
D is a simple vertical shift of 9.
C will represent the horizontal shift. Right now the curve is set 0,9. Since we need to shift to the left (x-C) and sin is involved, we need to do a little more math.
We want 11 = 2 sin (.49087(6-C)) + 9
11 = 2 sin (2.94522 - .49087C) + 9
Eliminate the 9, and cancel the 2s.
1 = sin (2.94522 - .49087C)
arcsin 1 (in radians) = 1.57079
1.57079 = 2.94522 - .49087C
.49087C = 2.94522 - 1.57079
.49087C = 1.37443
C = 2.8
The final equation is y=2sin (.49087(x-2.8))+9
Try this out on the link below to see for yourself.