https://file.aiquickdraw.com/mathbot/file/172544757913710xmlyhf.jpg

Here is the graph of the position function \( x(t) = 9.75 + 1.50t^3 \) for the time interval from \( t = 2.00 \) s to \( t = 3.00 \) s. 

### Graph Details:
- The blue curve represents the position \( x(t) \) as a function of time.
- The red points mark the positions at \( t = 2.00 \) s, \( t = 2.50 \) s, and \( t = 3.00 \) s.
- The green arrows indicate the instantaneous velocities at these points, calculated from the derivative \( v(t) = 4.50t^2 \).

### Summary of Results:
- **Average Velocity** over the interval \( t = 2.00 \) s to \( t = 3.00 \) s is 28.5 cm/s (as given).
- **Instantaneous Velocity**:
  - At \( t = 2.00 \) s: \( 18.00 \) cm/s
  - At \( t = 2.50 \) s: \( 28.12 \) cm/s
  - At \( t = 3.00 \) s: \( 40.50 \) cm/s

Would you like any more details or have any questions? Here are five related questions to explore further:

1. How would the graph change if the cubic coefficient were negative?
2. What is the significance of the slope of the position-time graph?
3. How do the instantaneous velocities compare to the average velocity?
4. What happens to the velocity as time increases beyond \( t = 3.00 \) s?
5. How would you find the exact time when the particle is at the midpoint between two given positions?

**Tip:** The slope of the position-time graph at any point represents the instantaneous velocity at that time.

The position of a particle moving along the x-axis is given in centimeters by x = 9.75 + 1.50t^3, where t is in seconds. Consider the time interval t = 2.00 s to t = 3.00 s and calculate: a) the average velocity, b) the instantaneous velocity at t = 2.00 s, c) the instantaneous velocity at t = 3.00 s, d) the instantaneous velocity at t = 2.50 s, e) the instantaneous velocity when the particle is midway between its positions at t = 2.00 s and t = 3.00 s, f) Graph x versus t and indicate your answers graphically.

Learn how to calculate the average and instantaneous velocities of a particle moving along the x-axis given by the position function x(t) = 9.75 + 1.50t^3. Explore concepts of calculus and kinematics, and visualize the results graphically. Suitable for high school students in Grades 11-12.

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