the vertical displacement y of a boat that is rocking up and down a lake, with y measured relative to the bottom of the lake. It has a maximum displacement of 12 meters and a minimum of 8 meters, a period of 3 seconds and an initial displacement of 11 meter when measurements was first started (i.e. t=0)

find the the trigonometric function to describe the rhythmic process.

the answer is y(t)=10+2sin(2pi/3(t+1/4))

I was wondering why the phase shift is +1/4 and not -1/4 . Can anyone explain this to me please?

the initial value of 11 means that

2sin(2pi/3(t+1/4)) = 1
sin(2pi/3(t+1/4)) = 1/2

2pi/3(t+1/4) = pi/6
t + 1/4 = 1/4
t=0

With no displacement, y(0) = 0
since sin(x) is a rising curve at x=0, the graph needs to be shifted left to bring the value of 1 into play.

Try playing around here with the values:

http://www.wolframalpha.com/input/?i=10%2B2sin(2pi%2F3(t%2B1%2F4))

Well, it seems like the boat really likes to party with a positive phase shift! But let me explain it in a more serious way.

The phase shift in a trigonometric function determines how the graph is shifted horizontally. In this case, the function is of the form sin(ωt+ϕ), where ω represents the angular frequency and ϕ represents the phase shift.

In the equation y(t) = 10 + 2sin(2π/3(t+1/4)), the phase shift is indeed positive, given by +1/4. This means that the graph of the function is shifted to the left by (1/4) of its period.

To understand why it's +1/4, we need to consider the initial displacement. At t=0, the initial displacement is given as 11 meters. So, we need to find the value of t when the function y(t) reaches its maximum displacement of 12 meters.

If we solve the equation y(t) = 10 + 2sin(2π/3(t+1/4)) = 12, we would find that t ≈ -1/4. However, since t represents time, which starts at t=0 in this case, we shift the value of t by +1/4 to bring it back to a positive value. This adjustment allows the function to start at the initial displacement of 11 meters at t=0.

So, in summary, the positive phase shift of +1/4 ensures that the function matches the initial displacement when t=0 and aligns with the given maximum and minimum displacements while maintaining the correct period of 3 seconds.

Hope that clears things up and gives you a chuckle too!

The phase shift in the trigonometric function describes the horizontal displacement of the waveform. In this case, the phase shift is given as +1/4, which means the waveform is shifted to the left by 1/4 of its period.

Let's break it down step by step to understand why the phase shift is positive instead of negative:

1. The period of the waveform is given as 3 seconds. This means the waveform completes one full cycle in 3 seconds.
2. Since the initial displacement at t=0 is 11 meters, we can use this information to find the equation for the vertical displacement.
3. We want the waveform to oscillate between a maximum displacement of 12 meters and a minimum of 8 meters. The average displacement between these two extremes is (12 + 8) / 2 = 10 meters.
4. The general equation for the vertical displacement of a sinusoidal waveform is y(t) = A + Bsin(2π/period * (t - phase shift)), where A is the average displacement and B is half of the amplitude (maximum displacement - average displacement).
5. Plugging in the values we know, the equation becomes y(t) = 10 + 2sin(2π/3(t - phase shift)).
6. We can find the phase shift by comparing the given equation with the general equation. In this case, the phase shift is the value that gives a displacement of 11 meters at t = 0.
When we substitute t = 0, the equation becomes 11 = 10 + 2sin(2π/3(- phase shift)).
7. Rearranging the equation, we get 1 = sin(2π/3(- phase shift)).
8. To solve for the phase shift, we need to find the angle whose sine is 1. The angle with a sine of 1 is π/2.
So, we have 2π/3(- phase shift) = π/2.
9. Solving for the phase shift, we get (- phase shift) = (π/2)/(2π/3) = π/4.
10. To get the positive phase shift, we multiply (- phase shift) by -1, which gives us (+ phase shift) = -π/4 = -0.25.
11. Therefore, the phase shift in the trigonometric function should be -0.25 rather than +0.25.

In conclusion, it seems there was a mistake in the given answer. The correct trigonometric function to describe the rhythmic process should be y(t) = 10 + 2sin(2π/3(t - 0.25)).

To understand why the phase shift is +1/4 and not -1/4, let's start by considering the general form of the trigonometric function describing the vertical displacement:

y(t) = A + Bsin(ωt + φ)

Where:
- A represents the vertical shift or the average displacement from the equilibrium position,
- B represents the amplitude or the maximum displacement from the average position,
- ω represents the angular frequency, given by ω = 2π/T (where T is the period),
- φ represents the phase shift, which indicates the horizontal shift of the graph.

In the given problem, we have the following information:
- Maximum displacement (amplitude), B = 12 meters
- Minimum displacement, A = 8 meters
- Period, T = 3 seconds (so angular frequency ω = 2π/3)
- Initial displacement at t = 0, A = 11 meters

Using these values, we can determine the average displacement A and the amplitude B:

A = (maximum displacement + minimum displacement) / 2
= (12 + 8) / 2
= 10 meters

B = (maximum displacement - minimum displacement) / 2
= (12 - 8) / 2
= 2 meters

Now, to find the phase shift φ, we can use the initial displacement A = 11 meters. Plugging these values into the trigonometric function, we have:

y(t) = 10 + 2sin(ωt + φ)

At t = 0, the function should equal the initial displacement, which is 11. So we can set up the equation:

11 = 10 + 2sin(ω(0) + φ)
11 = 10 + 2sin(φ)

To isolate φ, we subtract 10 from both sides:

2 = 2sin(φ)

Dividing both sides by 2, we get:

1 = sin(φ)

Now, to find the value of φ, we need to determine when sin(φ) equals 1. Since sin(φ) represents the vertical displacement and has a maximum value of 1, φ should be equal to a value that provides the maximum displacement of the function.

In this case, we know that when t = 0, the function should be at its maximum displacement, which corresponds to the initial point of measurement. So, sin(φ) = 1 when φ = 0.

Now, we rewrite the equation with the determined values:

y(t) = 10 + 2sin(2π/3(t + 0))

Simplifying further:

y(t) = 10 + 2sin(2π/3t)

So, the phase shift φ is considered to be 0 (or +0) because the initial measurement was taken at the maximum displacement. Therefore, the trigonometric function to describe the rhythmic process is:

y(t) = 10 + 2sin(2π/3t)