Write a sine and a cosine equation to model the movement of the boat in the following case: A boat tied at a dock moves up and down with the passing waves.  The vertical distance between its high point and low point is 9 m, and 5 cycles are repeated every minute.  Assume the motion can be modeled by a sinusoidal function, where the distance from the equilibrium position d(t), is measured in metres with respect to time in seconds.  At t=0, the boat is at its lowest point.

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The vertical distance between the high and the low points of the boat is 9 m. This means that the high point is 4.5 m above the mean and the low point is 4.5 m below the mean. And there are 5 cycles in a minute.

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The vertical distance between the high and the low points of the boat is 9 m. This means that the high point is 4.5 m above the mean and the low point is 4.5 m below the mean. And there are 5 cycles in a minute.

Taking time on the x-axis, and equating 1 unit on the x-axis to 1 minute, we can  model the vertical motion of the boat using a cosine function as: d(t) = -4.5 cos(10*pi*t). As a sin function it can be modeled as d(t) = -4.5 sin ( pi/2 - 10*pi*t)

We see that at t = 0, -4.5 cos(10*pi*t) = -4.5 sin ( pi/2 - 10*pi*t) = -4.5, which is the lowest distance. Also, there are 5 cycles per minutes, or one cycle in 360/10*pi = 0.2 seconds.

The required equations are d(t) = -4.5 cos(10*pi*t) and d(t) = -4.5 sin ( pi/2 - 10*pi*t).

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