Question

a boat that is tied to a dock moves vertically up and down with the waves. dieray watches the boat for 30 seconds and note that the boat moves up and down a total 6 times. the difference between the boat's highest point and lowest point is 3 feet. write and graph a trigonometric function that models the boat's vertical position x seconds after the began watching. assume that when dieray began watching the boat, it was at its highest point and that its average vertical position was 0 feet.

Answer 1

The motion of the boat can be modeled by the trigonometric function f(x) = 1.5 sin(π/5 x), representing the boat's vertical position in feet x seconds after observation begins.

Step-by-step explanation:

The motion of the boat moving up and down with the waves can be modeled using a trigonometric function, specifically a sine function, because we are given that the boat returns to its highest point every 5 seconds, which is the period of the motion since it goes up and down 6 times in 30 seconds. Since the boat starts at the highest point, the phase shift is 0. The amplitude of the motion is 1.5 feet as this is half the total vertical distance between the highest and lowest points. We can write the function as:

f(x) = A sin(B(x - C)) + D

Where A is the amplitude, B is related to the period, C is the phase shift, and D is the vertical shift. Using the information given, we get:

f(x) = 1.5 sin(π/5 x)

This function describes the boat's vertical position, in feet, x seconds after Dieray began watching. To graph this function, plot the sine wave starting at the point (0,1.5), going down to (5,-1.5), and repeating this cycle every 10 seconds.