A pendulum is swinging next to a wall.

The distance \[D(t)\] (in \[\text{cm}\]) between the bob of the pendulum and the wall as a function of time \[t\] (in seconds) can be modeled by a sinusoidal expression of the form \[a\cdot\sin(b\cdot t)+d\].

At \[t=0\], when the pendulum is exactly in the middle of its swing, the bob is \[5\text{ cm}\] away from the wall. The bob reaches the closest point to the wall, which is \[3\text{ cm}\] from the wall, \[1\] second later.

**Find \[D(t)\].**
*\[\text{\it{t}}\] should be in radians.*

\[D(t) = \] 

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To model the distance \( D(t) \) between the bob of the pendulum and the wall, we use the sinusoidal function in the form:

\[
D(t) = a \cdot \sin(b \cdot t) + d
\]

### Step-by-Step Solution:

1. **Determine the parameters \( a, b, \) and \( d \):**

   - **Amplitude \( a \):**  
     The amplitude of a sinusoidal function represents half the difference between the maximum and minimum values.  
     The bob reaches the closest point to the wall, \(3 \, \text{cm}\), and at the middle of the swing, it is \(5 \, \text{cm}\) away.  
     Since this is at \( t = 0 \) and corresponds to the mean position in the sinusoid, we set this as the sinusoid's midline.

   - **Vertical shift \( d \):**  
     The vertical shift \( d \) is the mean value of the maximum and minimum distances of the pendulum from the wall.  
     Since the middle distance is given directly as \(5 \, \text{cm}\), we set \( d = 5 \).

   - **Amplitude \( a \):**  
     The minimum distance \(3 \, \text{cm}\) from the wall tells us that the amplitude is the distance from the midline to the minimum point:
     \[
     a = d - 3 = 5 - 3 = 2.
     \]

   - **Period and Frequency \( b \):**  
     The pendulum reaches the closest point at \( t = 1 \) second, which indicates a quarter-period of the sinusoid.  
     If \( T \) is the period, then \( \frac{T}{4} = 1 \) second, leading to \( T = 4 \) seconds.  
     Since the formula for period in terms of \( b \) is \( T = \frac{2\pi}{b} \), we can solve for \( b \):
     \[
     \frac{2\pi}{b} = 4 \implies b = \frac{2\pi}{4} = \frac{\pi}{2}.
     \]

2. **Construct the function \( D(t) \):**

   Now we substitute these values into the general form:
   \[
   D(t) = 2 \cdot \sin\left( \frac{\pi}{2} \cdot t \right) + 5.
   \]

### Final Answer:

\[
\boxed{D(t) = 2 \sin\left( \frac{\pi}{2} t \right) + 5}
\]

This is the function that models the distance \( D(t) \) of the pendulum bob from the wall over time in seconds with \( t \) in radians. 

Would you like to see more details or ask any questions?

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### Follow-up Questions:

1. What would happen to the function \( D(t) \) if the amplitude were doubled?
2. How does the period of the pendulum affect the parameter \( b \) in the sine function?
3. What is the effect of changing the vertical shift \( d \) on the graph of \( D(t) \)?
4. How would the function change if the closest distance to the wall were \(2 \, \text{cm}\) instead of \(3 \, \text{cm}\)?
5. If the period of the pendulum were \(6\) seconds, how would you adjust the parameter \( b \)?

### Tip:
To visualize sinusoidal functions, try graphing them to see how changing each parameter affects the shape of the curve.

Learn how to model the distance of a pendulum bob from a wall over time using a sinusoidal function. Understand the parameters a, b, and d in the function D(t) = a * sin(b * t) + d and their significance. Includes step-by-step solution and examples. Suitable for high school students studying trigonometry and sinusoidal functions.

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