# Meagan is sitting in a rocking chair, the distance d(t-) between the wall and the rear of the chair varies sinusoidally with time t. At t = 1s , the chair is closest to the wall and d(1) = 18 cm. At t = 1.75s the chair is farthest from the wall and d(1.75 ) = 34m

how far is the chair when no one is rocking it?

and

what is the distance between wall and the chair at t= 8s?

## Let's use the equation d(t) = A * sin(B * (t - C)) + D. The amplitude A is equal to half the difference between the maximum and minimum distances (34 cm - 18 cm) / 2 = 16 cm / 2 = 8 cm. The center of oscillation D is the average of the maximum and minimum distances, (34 cm + 18 cm) / 2 = 52 cm / 2 = 26 cm.

Now we can find the phase shift C. Since Meagan is closest to the wall at t = 1s, we have sin(B * (1 - C)) = -1, because the sine function reaches its minimum at a 270 degrees or 3π/2 radians, we can write the equation as:

B * (1 - C) = 3π/2

We can also find the period of the oscillation using the information given about the chair at t = 1.75s. Since the chair is farthest from the wall at t = 1.75s, we have sin(B * (1.75 - C)) = 1. We know that the sine function reaches its maximum at 90 degrees or π/2 radians, so we can write the equation as:

B * (1.75 - C) = π/2

Since both equations equal the sine function's maximum and minimum, respectively, we can set them equal to each other:

3π/2 * (1 - C) = π/2 * (1.75 - C)
3(1 - C) = 1.75 - C
C = 2/7

Now we can find B:

B * (1 - 2/7) = 3π/2
B * (5/7) = 3π/2
B = (3π/2) * (7/5)
B = 3π/2 * 7/5

We're now ready to find the distance between the wall and the chair when no one is rocking it, which is at t = 0:

d(0) = 8 * sin(3π/2 * 7/5 * (0 - 2/7)) + 26
d(0) = 8 * sin(-6π/5) + 26
d(0) ≈ 8 * 0.951 + 26
d(0) ≈ 7.608 + 26
d(0) ≈ 33.608 cm (approximately)

So when no one is rocking the chair, the distance between the wall and the chair is about 33.608 cm.

Now let's find the distance between the wall and the chair at t = 8s:

d(8) = 8 * sin(3π/2 * 7/5 * (8 - 2/7)) + 26
d(8) ≈ 8 * sin(0.760) + 26
d(8) ≈ 8 * 0.686 + 26
d(8) ≈ 5.488 + 26
d(8) ≈ 31.488 cm (approximately)

So the distance between the wall and the chair at t = 8s is about 31.488 cm.

## To determine the distance when no one is rocking the chair, we need to find the average distance between the wall and the rear of the chair over one complete cycle of the sine function. We are given that at t = 1s, the chair is closest to the wall with d(1) = 18 cm, and at t = 1.75s, the chair is farthest from the wall with d(1.75) = 34 cm.

First, we need to find the time period of the sinusoidal motion. The time it takes for one complete cycle can be calculated by subtracting the time at the closest position from the time at the farthest position:

Time period (T) = t(farthest position) - t(closest position)
= 1.75s - 1s
= 0.75s

Now, we can find the average distance by finding the midpoint between the closest and farthest positions:

Average distance = (d(closest) + d(farthest)) / 2
= (18 cm + 34 cm) / 2
= 52 cm / 2
= 26 cm

Therefore, when no one is rocking the chair, the distance between the wall and the chair is 26 cm.

Now, let's find the distance between the wall and the chair at t = 8s. Since the distance varies sinusoidally with time, we can use the sine function to represent this:

d(t) = A*sin(2π/T * (t - t0)) + C

Where:
A = amplitude of the oscillation (distance between the closest and farthest positions)
T = time period of the oscillation
t0 = phase shift (time at the closest position)
C = vertical shift (average distance when no one is rocking the chair)

Given values:
A = (34 cm - 18 cm) / 2 = 8 cm
T = 0.75s
t0 = 1s
C = 26 cm

Substituting the values into the equation:

d(t) = 8*sin(2π/0.75 * (t - 1)) + 26

Now, let's find the distance at t = 8s:

d(8) = 8*sin(2π/0.75 * (8 - 1)) + 26

Calculating this expression gives us the distance between the wall and the chair at t = 8s.

## To find the distance when no one is rocking the chair, we need to determine the amplitude of the sinusoidal function. The amplitude can be calculated by finding the average of the maximum and minimum values of the function.

Given that at t = 1s, the chair is closest to the wall with a distance of d(1) = 18 cm,
and at t = 1.75s, the chair is farthest from the wall with a distance of d(1.75) = 34 cm,

The average of these two values will give us the amplitude:
Amplitude = (18 cm + 34 cm) / 2
Amplitude = 26 cm

Therefore, when no one is rocking the chair, the distance will be equal to the amplitude, which is 26 cm.

To find the distance between the wall and the chair at t = 8s, we need to determine the equation of the sinusoidal function.

The general equation for a sinusoidal function is:
d(t) = A * sin(ωt + φ) + C

Where:
- A is the amplitude (already determined as 26 cm)
- ω is the angular frequency (the number of complete cycles per unit of time)
- φ is the phase shift (the horizontal displacement of the sinusoidal function)
- C is the vertical displacement (the mean or average value of the function)

Since we don't have information about the angular frequency or phase shift, we cannot determine the exact equation of the function. Therefore, we cannot find the distance at t = 8s without additional information.