Question

1. a boat is bobbing up and down on the water. the distance between the boats highest and lowest points is 4 m. the boat moves from its highest point to its lowest point and back to its highest point every 30 s. write a cosine function that models the movement of the boat in relation to the equilibrium point.

Explanation:

Step1: Determine the amplitude

The distance between the highest and lowest points is \(4\) m. The amplitude \(A\) is half of this distance. So, \(A=\frac{4}{2} = 2\) m.

Step2: Determine the period

The period \(T\) is the time it takes for the boat to move from its highest point to its lowest point and back to its highest point. Given \(T = 30\) s. The formula for the angular frequency \(\omega\) is \(\omega=\frac{2\pi}{T}\). Substituting \(T = 30\), we get \(\omega=\frac{2\pi}{30}=\frac{\pi}{15}\).

Step3: Write the cosine function

Assuming the boat starts at its highest point (so there is no phase shift \(C = 0\) and no vertical shift \(D=0\)), the general form of a cosine function is \(y = A\cos(\omega x)+D\). Substituting \(A = 2\), \(\omega=\frac{\pi}{15}\) and \(D = 0\), we get \(y=2\cos(\frac{\pi}{15}x)\)

Answer:

\(y = 2\cos(\frac{\pi}{15}x)\)