a buoy is placed at a depth of 10 meters below the surface of the water. the buoy bobs up and then down 6…

a buoy is placed at a depth of 10 meters below the surface of the water. the buoy bobs up and then down 6 meters from its original placement depth. ten seconds pass from the time the buoy is at its highest point to when it is at its lowest point. assume at x = 0 the buoy is at its original placement depth. let y = 0 represent the surface of the water. use the sine tool to graph the function. the first point must be on the mid - line and the second point must be a maximum or minimum value on the graph closest to the first point.

a buoy is placed at a depth of 10 meters below the surface of the water. the buoy bobs up and then down 6 meters from its original placement depth. ten seconds pass from the time the buoy is at its highest point to when it is at its lowest point. assume at x = 0 the buoy is at its original placement depth. let y = 0 represent the surface of the water. use the sine tool to graph the function. the first point must be on the mid - line and the second point must be a maximum or minimum value on the graph closest to the first point.

Answer

Explanation:

Step1: Determine the mid - line

The buoy is initially at a depth of 10 meters below the water surface ((y = 0) represents the water surface), so the mid - line of the sine function is (y=- 10).

Step2: Determine the amplitude

The buoy bobs up and down 6 meters from its original placement depth, so the amplitude (A = 6).

Step3: Determine the period

It takes 10 seconds from the highest point to the lowest point. The period (T) of a sine function is the time from one highest (or lowest) point to the next highest (or lowest) point. Since the time from highest to lowest is half of the period, (T = 20) seconds. Then the angular frequency (\omega=\frac{2\pi}{T}=\frac{2\pi}{20}=\frac{\pi}{10}).

Step4: Write the sine function

The general form of a sine function is (y = A\sin(\omega x+\varphi)+k). Since at (x = 0) the buoy is at the mid - line, and we want a sine function, (\varphi = 0) and (k=-10), (A = 6), (\omega=\frac{\pi}{10}), so the function is (y=-10 + 6\sin(\frac{\pi}{10}x)).

Step5: Find the first and second points

For the first point on the mid - line, when (x = 0), (y=-10). For the second point (a maximum or minimum closest to the first point), for a sine function (y=-10 + 6\sin(\frac{\pi}{10}x)), the maximum value occurs when (\sin(\frac{\pi}{10}x)=1), (\frac{\pi}{10}x=\frac{\pi}{2}), (x = 5) and (y=-10 + 6=-4) (a maximum).

To graph using the sine - tool:

  1. Place the first point at ((0,-10)) (on the mid - line (y=-10)).
  2. Place the second point at ((5,-4)) (a maximum point of the function (y=-10 + 6\sin(\frac{\pi}{10}x))).

Answer:

First point: ((0,-10)), Second point: ((5,-4))