Jordan is sitting in the walking chair. the distance d(t), between the wall and the rear of the chair varies sinusoidally with time t. at t=1 is the chair closest to the wall and d(1)=1.5m. At t=2 the chair is the farthest from the wall and d(2)= 3.2m
What is the distance between the wall and the chair at t=8s

To solve the problem, we need to model the distance \( d(t) \) as a sinusoidal function. A sinusoidal function can be generally represented as:

\[ d(t) = A \sin(B(t - C)) + D \]

where:
- \( A \) is the amplitude,
- \( B \) is related to the period of the function,
- \( C \) is the phase shift, and
- \( D \) is the vertical shift.

Given the information:
1. \( d(1) = 1.5 \) meters (closest to the wall),
2. \( d(2) = 3.2 \) meters (farthest from the wall).

### Step 1: Determine the vertical shift \( D \)
The average of the maximum and minimum values gives us \( D \):

\[ D = \frac{1.5 + 3.2}{2} = \frac{4.7}{2} = 2.35 \]

### Step 2: Determine the amplitude \( A \)
The amplitude \( A \) is half the distance between the maximum and minimum values:

\[ A = \frac{3.2 - 1.5}{2} = \frac{1.7}{2} = 0.85 \]

### Step 3: Determine the period and \( B \)
The period \( T \) is the time it takes for the function to complete one full cycle. Here, since the chair is closest to the wall at \( t = 1 \) and farthest at \( t = 2 \), half the period is from the closest point to the farthest point. Therefore, a full period would be:

\[ T = 2 \times (2 - 1) = 2 \text{ seconds} \]

Since the period \( T \) is related to \( B \) by \( T = \frac{2\pi}{B} \):

\[ 2 = \frac{2\pi}{B} \implies B = \pi \]

### Step 4: Determine the phase shift \( C \)
Since the chair is closest to the wall at \( t = 1 \) and \( \sin \) reaches its minimum at \( \frac{3\pi}{2} \), we can set:

\[ B(t - C) = \frac{3\pi}{2} \]
\[ \pi(1 - C) = \frac{3\pi}{2} \]
\[ 1 - C = \frac{3}{2} \]
\[ C = 1 - \frac{3}{2} = -\frac{1}{2} \]

Thus, the equation of the distance function is:

\[ d(t) = 0.85 \sin(\pi(t + \frac{1}{2})) + 2.35 \]

### Step 5: Find \( d(8) \)
Now we substitute \( t = 8 \) into the equation:

\[ d(8) = 0.85 \sin\left(\pi \left(8 + \frac{1}{2}\right)\right) + 2.35 \]
\[ d(8) = 0.85 \sin\left(\pi \cdot 8.5\right) + 2.35 \]
\[ d(8) = 0.85 \sin(8.5\pi) + 2.35 \]

Since \( 8.5\pi \) is an odd multiple of \( \pi \), the sine function evaluates to \( -1 \):

\[ \sin(8.5\pi) = \sin\left(\frac{17\pi}{2}\right) = -1 \]

Thus:

\[ d(8) = 0.85 \cdot (-1) + 2.35 \]
\[ d(8) = -0.85 + 2.35 \]
\[ d(8) = 1.5 \text{ meters} \]

So, the distance between the wall and the chair at \( t = 8 \) seconds is 1.5 meters.

Would you like more details or have any questions?

Here are some questions you might find interesting:
1. How do we determine the vertical shift in a sinusoidal function?
2. What is the significance of the amplitude in a sinusoidal function?
3. How can we find the period of a sinusoidal function from given data points?
4. What is the phase shift and how is it determined in this context?
5. How do sinusoidal functions model real-world periodic phenomena?

**Tip:** When working with sinusoidal functions, remember that the amplitude represents the maximum deviation from the average value, and the period indicates how long it takes for the function to complete one cycle.

Explore how to calculate the distance between Jordan's walking chair and the wall at different times using sinusoidal functions. Learn step-by-step with a detailed solution, including determining amplitude, period, phase shift, and vertical shift. Suitable for high school students studying trigonometry and sinusoidal functions.

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