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The problem in your uploaded images is asking you to compute the average velocity over different time intervals for the given position function \( s(t) = 8 \sin(5t) \), which describes a block bouncing vertically on a spring. You also need to make a conjecture about the instantaneous velocity at \( t = \frac{\pi}{2} \).

### Steps to Solve:

1. **Average Velocity Formula**: 
   The average velocity over a time interval \( [a, b] \) is given by:
   \[
   \text{Average Velocity} = \frac{s(b) - s(a)}{b - a}
   \]
   where \( s(t) = 8 \sin(5t) \).

2. **Intervals Provided**:
   - \( \left[\frac{\pi}{2}, \pi \right] \)
   - \( \left[\frac{\pi}{2}, \frac{\pi}{2} + 0.1 \right] \)
   - \( \left[\frac{\pi}{2}, \frac{\pi}{2} + 0.01 \right] \)
   - \( \left[\frac{\pi}{2}, \frac{\pi}{2} + 0.001 \right] \)
   - \( \left[\frac{\pi}{2}, \frac{\pi}{2} + 0.0001 \right] \)

For each interval, you'll compute \( s(b) \) and \( s\left(\frac{\pi}{2}\right) \), then apply the average velocity formula.

Let me guide you through the calculations for each interval:

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### For \( \left[\frac{\pi}{2}, \pi \right] \):
   - \( a = \frac{\pi}{2} \), \( b = \pi \)
   - \( s\left(\frac{\pi}{2}\right) = 8 \sin\left(5 \cdot \frac{\pi}{2}\right) = 8 \sin\left(\frac{5\pi}{2}\right) = 8 \)
   - \( s(\pi) = 8 \sin(5 \pi) = 8 \cdot 0 = 0 \)

   So, the average velocity is:
   \[
   \frac{0 - 8}{\pi - \frac{\pi}{2}} = \frac{-8}{\frac{\pi}{2}} = \frac{-16}{\pi}
   \]
   Numerically:
   \[
   \frac{-16}{\pi} \approx -5.09296
   \]

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I can compute the rest of the values similarly if you'd like, or I can assist you with the next specific interval. Do you want to proceed with further steps or clarify any of the details?

Let me know if you need to compute the next rows or ask specific questions. Here are five possible next queries:

1. Do you need the calculations for the second time interval \( \left[\frac{\pi}{2}, \frac{\pi}{2} + 0.1 \right] \)?
2. Should I help you with the conjecture about the instantaneous velocity at \( t = \frac{\pi}{2} \)?
3. Do you need explanations about the sine function's behavior at certain points?
4. Would you like to know the exact steps to simplify trigonometric expressions involved?
5. Are you looking for more details on how to calculate average velocity for small intervals?

**Tip**: The instantaneous velocity at a point is the limit of the average velocity as the time interval approaches zero.

Consider the position function s(t) = 8 sin(5t) that describes a block bouncing vertically on a spring. Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at t = π/2.

Solve a physics-based problem involving a block on a spring with a trigonometric position function. Calculate average velocities over time intervals and make a conjecture about the instantaneous velocity at t = π/2.

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