boat tied up at a dock bobs up and down with passing waves. The vertical distance between its high point and its low point is 1.8m and the cycle is repeated every 4 seconds.
a) Determine a sinusoidal equation to model the vertical position, in metres, of the boat versus the time, in seconds.
b) Use your model to determine when, during each cycle, the boat is 0.5m above its mean position. Round your answers to the nearest hundredth of a second.
y=0.9sin(pie/2t)
So clearly the amplitude is .9 m
2π/k = period
2π/k = 4
4k = 2π
k = π/2
but we want the low point to be 0 not -.9
so height = .9sin(π/2 t), where t is in seconds and height is in metres
Reiny, you are not right. The textbook answers say otherwise. 😡😡😡
a) Well, I can certainly "wave" some humor into this question! Let's get rockin' and rollin' with this sinusoidal party.
The general equation for a sinusoidal function is given by:
y = A * sin(B * (x - C)) + D
Where A is the amplitude, B is the frequency, C is the phase shift, and D is the vertical shift.
In this case, we know that the vertical distance between the high point and the low point is 1.8m, so the amplitude, A, would be half of that, which is 0.9m.
The cycle is repeated every 4 seconds, so the frequency, B, would be 2π/4 = π/2.
There is no phase shift in this problem, so C = 0.
Since the high point is the mean position, the vertical shift, D, would be 0.
Putting it all together, our sinusoidal equation to model the vertical position of the boat would be:
y = 0.9 * sin(π/2 * x)
b) Now let's determine when the boat is 0.5m above its mean position. To do this, we can set up an equation:
0.5 = 0.9 * sin(π/2 * x)
Now, I'm not a math magician, but I know a trick or two. Let's solve for x by taking the inverse sin of both sides:
sin^(-1)(0.5) = π/2 * x
Using our trusty calculators, we get:
x ≈ 0.64 seconds
So, the boat is approximately 0.5m above its mean position at 0.64 seconds during each cycle.
Hope that waves away any confusion and keeps you afloat with a smile!
To determine a sinusoidal equation that models the vertical position of the boat, we can use the equation of a sine function.
The general equation for a sine function is: y = A*sin(B(x - C)) + D
Where:
A is the amplitude of the function (half the vertical distance between the high and low points),
B determines the frequency of the function (number of cycles in a given period),
C represents the phase shift or horizontal displacement, and
D is the vertical displacement or mean position of the graph.
In this case, the amplitude is half of the vertical distance between the high and low points, which is 1.8m/2 = 0.9m. The frequency is 1 cycle every 4 seconds, which corresponds to a period of 4 seconds. Therefore, B = 2π/4 = π/2.
The phase shift depends on when we start measuring time. Assuming time = 0 corresponds to the boat at its highest point, it means the time taken for the boat to reach its lowest point is half of the period, which gives a phase shift of C = 4s/2 = 2s.
Since the boat is tied up at the dock, its mean position is 0. Therefore, D = 0.
Putting these values into the equation, we get:
y = 0.9*sin((π/2)(x - 2))
Now let's use this model to determine when the boat is 0.5m above its mean position.
Setting y = 0.5 in the equation, we get:
0.5 = 0.9*sin((π/2)(x - 2))
To solve for x, we need to isolate the sine function. Divide both sides by 0.9:
0.55555555555 = sin((π/2)(x - 2))
Now we can take the inverse sine (arcsin) of both sides to find the angle:
arcsin(0.55555555555) = (π/2)(x - 2)
Using a calculator, arcsin(0.55555555555) ≈ 0.58904862254 radians.
Now we can solve for x:
(π/2)(x - 2) ≈ 0.58904862254
Divide both sides by π/2:
x - 2 ≈ 0.58904862254 / (π/2)
x - 2 ≈ 0.37356791651
Add 2 to both sides:
x ≈ 2 + 0.37356791651
x ≈ 2.37356791651
Rounding to the nearest hundredth of a second, the boat is 0.5m above its mean position at approximately 2.37 seconds during each cycle.
Therefore, during each cycle, the boat is 0.5m above its mean position at approximately 2.37 seconds.
oops, pressed "submit" too soon , forgot to raise that boat:
height = .9sin(π/2 t) + .9
b) .9sin(π/2 t) + .9 > ?? ( is the normal position the water level, that is , zero
or is it 0.9 ? You decide, you will get 2 answers.)