Hello my dudes and gals I am stuck on a buoy question that my teacher game me hoping y'all can help. its a 2 parter
A buoy out in the ocean bobs up and down with the passing waves. The vertical distance between its high and low points is 3.2 m and the cycle is repeated every 8 secounds.
A) determine a sinusoidal equation to model the veritical position, in meters of the boat verse time, in secounds. Assume the buoy is at equillibirum/ average sea level at t=0.
B) find the time during the first cycle that the buoy is 1m above equillibirum/ average sea level, rounded 2 decimal places.
pls respond, if you do thanks and stay safe!
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
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