Another garbage buoy question

We are given that a boat bobs up and down with the waves such that the distance from highest to lowest point is 1.8 meters and the cycle is repeated every 4 seconds. We are asked to find the time(s) that the instantaneous rate of change of the vertical displacement is zero and when it achieves its maximum.

The underlying model is a sinusoid. Since there is no indication as to where or when to start, we can use a sine wave such that at t=0 the boat is midway between its highest and lowest point. We can also place the midline at y=0 as sea level.

The amplitude is .9 meters. (The boat goes up .9 meters from the midline and down .9 meters from the midline for a distance between maximum and minimum of 1.8 meters.) The period is 4 seconds (the time to repeat the action).

The function model is with where a is the amplitude and b is 2pi divided by the period. (See image for the graph.)

To determine the time when the instantaneous rate of change is zero we take the...

You should be able to take it from here, hope this helps, whymenotsmart^m^.

A) We have  the form

y  = A sin (Bt)

A = the amplitude = 3.2

We can  find B thusly

When  t  =  2,  y  =  3.2

So

3.2  =3.2sin (2B)

1 = sin (2B)

B = pi/4

So  we have

y = 3.2 sin  (pi t / 4)

Here's the graph : https://www.desmos.com/calculator/lndmanq4ij

B)   We want to solve this

1 = 3.2sin (pi t / 4)

(1/3.2)  = sin (pi t/4)

(5/16)  =  sin (pi t  / 4)

arcsin (5/16)  =  pi  t / 4

.3178  =  pi t  /4

.3178 ( 4/pi)  = t ≈  .4046   ≈  .40  seconds