Question 1202197: Out at sea a fishing boat rises and falls due to the stormy waves. The model h(t) = sin (36t) represents the displacement of the fishing boar, h(t), in metres at t seconds.
a) Calculate the period.
b) Determine the displacement at 30 s
c) Determine the displacement at 12 s
d) At what time, to the nearest tenth of a second, does the displacement first reach 0.7 m.
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Part (a)
I'll use x in place of t, and y in place of h(t).
The given equation is y = sin(36x)
Compare it to the template y = A*sin(B(x-C))+D
The variables are:
|A| = amplitude
B = helps determine the period, more on that later
C = phase shift, which is the side to side shift
D = up and down shift, useful to determine midline
In this case,
A = 1
B = 36
C = 0
D = 0
When in radian mode, the period T is found through this formula
T = 2pi/B
T = 2pi/36
T = pi/18
Again, this only applies when working in radian mode.
If you are in degree mode, then the period would be
T = 360/B
T = 360/36
T = 10
Since this result is much nicer, I have a feeling your teacher probably wants you to use degree mode.
However, be sure to ask for clarification.
The period represents how long each cycle takes. So the wave pattern repeats itself every 10 seconds.
Answer: The period is 10 seconds, assuming you are in degree mode (otherwise the period is pi/18 seconds in radian mode).
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Part (b)
I've replaced t with x.
I've replaced h(t) with y.
x = time in seconds
y = displacement, specifically how far up or down the boat is compared to the midline
For the rest of this homework, I'll assume your teacher wants degree mode.
Plug x = 30 into the equation to determine y.
y = sin(36x)
y = sin(36*30)
y = sin(1080)
y = 0
The displacement is 0.
A displacement of 0 means the boat is at the midline.
It is very likely the "midline" refers to the average sea level.
Note that 1080 is a multiple of 360. It means angles 1080 degrees and 360 degrees are coterminal.
Answer: 0 meters
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Part (c)
Follow a similar set of steps we did for part (b).
This time plug in x = 12.
y = sin(36x)
y = sin(36*12)
y = sin(432)
y = 0.9510565
This represents how high the boat is compared to the midline.
If the displacement was negative, then the boat would be below the midline.
Answer: Approximately 0.9510565 meters above the midline
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Part (d)
The previous two parts had us plug in an x value to find y.
This time we'll start with a known y value to find x.
There will be infinitely many solutions if we do not restrict the domain (because sine is not one-to-one).
However, we'll be looking for the smallest positive x solution, which represents the first occurrence.
We'll need the inverse sine, aka arcsine, to isolate x.
Your calculator likely shows this as a button labeled 
y = sin(36x)
0.7 = sin(36x)
sin(36x) = 0.7
36x = arcsin(0.7)+360*n or 36x = 180-arcsin(0.7)+360*n
36x = 44.427004+360*n or 36x = 180-44.427004+360*n
36x = 44.427004+360*n or 36x = 135.572996+360*n
x = (44.427004+360*n)/36 or x = (135.572996+360*n)/36
That represents the set of all possible solutions for x.
n is any integer.
The decimal values are approximate.
Let,
A = (44.427004+360*n)/36
B = (135.572996+360*n)/36
Here's a table of various integer n values, along with A and B as well.
| n | A | B | | -2 | -18.8 | -16.2 | | -1 | -8.8 | -6.2 | | 0 | 1.2 | 3.8 | | 1 | 11.2 | 13.8 | | 2 | 21.2 | 23.8 | | 3 | 31.2 | 33.8 |
Each decimal value is approximate.
Highlighted in red is the solution we're after.
This is the smallest positive x solution that makes sin(36x) = 0.7 true when in degree mode.
I recommend using a graphing calculator such as GeoGebra or Desmos to verify the answer is correct.
Here is a link to the interactive Desmos graph
https://www.desmos.com/calculator/oaieombfdm
The sine wave crosses the horizontal line at approximately (1.2, 0.7) to visually confirm x = 1.2 is the smallest positive x solution.
Answer: 1.2 seconds
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