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.

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).**

We know that any periodic motion is a simple harmonic motion (SHM) and could be written as x(t) = A sin (wt+p), where A is the amplitude , w is the angular velocity, p is the phase difference and t is the time in minutes.

Since the vertical maximum distance between the highest and lowest points in the motion of the boat is 9 m, the amplitude A = 9/2 = 4.5 m

So here, A = 9/2, w = 2pi/ 5 radians per minute.

So at the time t = 0, the equation of motion given by : -4.5 = 4.5 sin {5t+p}. So sin (5*0+p) = -1.

=> 5t+p = -pi/2, or p = -pi/2.

Therefore the required model of the motion is x(t) = 4.5 sin(5t-pi/2) in terms of sin function.

To write the equation in terms of Cosine function:

Since cos x = sin (pi/2 - x), we can rewrite the above equation as below:

x(t) = 4.5 cos {pi/2 - ( 5t-pi/2)}.

=> x(t) = 4.5 cos {-5wt + pi}.

=> x(t) = 4.5 cos (5wt- pi) , as cos (-x) = cosx.

Therefore the required equation of the simple harmonic motion of the boat is given by:

x(t) = 4.5 (5wt-pi). Or x (t) = cos (5t+pi) , as x(t) = x(t+2pi) for any SHM.

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