Suppose you were at the beach and notice that at 1:00 pm the tide is in, that is, the depth of water is at its deepest. At that time you find that the depth of the water at the end of the pier is 1.9 meters. At 7:00 that evening when the tide is out, the depth of the water is 1.1 meters. Assuming that the depth of the water varies sinusoidally with time, find an equation that will model the depth of the water.
A. y=1.5cos(pi/3(x-7)) + 0.4
B. y=1.5cos(pi/6(x-7)) + 0.4
C. y=0.4cos(pi/6(x-7)) + 1.5
D. y=-0.4cos(pi/6(x-7)) + 1.5
half-period is 6 hours, so a full period is 12 hours. So, if 2pi/k = 12, k = pi/6.
The max-min=1.9-1.1 = 0.8, so the amplitude is 0.4
So, the choice is C or D.
The midpoint is (1.9+1.1)/2 = 1.5, so we're still looking at C or D.
Since we apparently want a cosine function, whose minimum is at x=7, that would make it (D) since (C) has its max at x=7.
To find the equation that models the depth of the water, we can use the general form of a sinusoidal function:
y = A*cos(B(x-C)) + D
where A represents the amplitude, B represents the period, C represents the phase shift, and D represents the vertical shift.
We are given two data points:
At 1:00 pm (13:00), the depth of water is 1.9 meters.
At 7:00 pm (19:00), the depth of water is 1.1 meters.
First, let's determine the amplitude, A. The amplitude is half the difference between the highest and lowest points on the graph. In this case, A = (1.9 - 1.1)/2 = 0.4.
Next, let's determine the period, B. The period is the time it takes for the graph to complete one full cycle. In this case, since the tide goes from being in to out in 6 hours, the period is 6 hours. To convert this to radians, we multiply by π/12, since there are 12 hours in a full cycle: B = π/12.
The phase shift, C, represents the horizontal shift of the graph. In this case, since the tide is in at 1:00 pm and out at 7:00 pm, there is a 6-hour phase shift to the right. So C = 7.
Lastly, the vertical shift, D, is the average of the highest and lowest points on the graph. In this case, D = (1.9 + 1.1)/2 = 1.5.
Plugging these values into the general form of the equation, we get:
y = 0.4*cos((π/12)(x-7)) + 1.5
Simplifying this equation gives the answer as:
Option C: y = 0.4*cos((π/6)(x-7)) + 1.5