Question

A boat sits on top of the water and rises and falls as the waves move through the ocean. 7 seconds after you spotted the boat it was at its lowest point (44 feet above the ocean floor) and 19 seconds after you spotted the boat it was at its highest point (56 feet above the ocean floor). Assume the boat’s distance above the ocean floor in terms of the number of seconds since you spotted the boat follows a sinusoidal model. Write the model for the boat’s distance above the ocean floor (in feet) in terms of x, the number of seconds since you first spotted the boat. Use the model H(x) = sin(x − phi) + or H(x) = sin((x − )) + .

Answer 1

The sinusoidal model for the boat's distance above the ocean floor as a function of time since it was first spotted is H(x) = 6 sin((2π/24)(x - 7)) + 50, where x is the number of seconds passed.

Step-by-step explanation:

The question requires the creation of a sinusoidal model to describe the simple harmonic motion of a boat's vertical position above the ocean floor over time. The model will be based on information about the boat's lowest and highest points above the ocean floor and the time it took to reach those points.

Let's find the amplitude, vertical shift, and phase shift for the sinusoidal function. The amplitude (A) is half the distance between the highest and lowest points. The vertical shift (D) is the average of the highest and lowest points, and the phase shift (phi) is determined by the time it took to reach the first extreme point from the start.

The time between the lowest and highest point is 19 - 7 = 12 seconds. Since these points are half a cycle apart, the period (T) is twice this amount, 24 seconds.

The amplitude is (56 - 44)/2 = 6 feet. The vertical shift is (56 + 44)/2 = 50 feet. The phase shift is 7 seconds because the lowest point was observed 7 seconds after we started timing. Using these values, the sinusoidal function model is:

H(x) = 6 sin((2π/24)(x - 7)) + 50

This equation models the boat's distance above the ocean floor as a function of x, the number of seconds since the boat was first spotted.