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Let's go step by step to solve the given questions based on the Ferris wheel graph:

### a) **Calculate the average rate of change in height on the interval [0, 4]:**

The average rate of change for a function over an interval \([a, b]\) is given by the formula:

\[
\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}
\]

- For the interval \([0, 4]\), the time is from 0 to 4 seconds.
- From the graph:
  - At \( t = 0 \), the height \( f(0) = 0 \) meters.
  - At \( t = 4 \), the height \( f(4) = 6 \) meters.

So, the average rate of change is:

\[
\frac{f(4) - f(0)}{4 - 0} = \frac{6 - 0}{4 - 0} = \frac{6}{4} = 1.5 \, \text{meters per second}.
\]

### b) **Calculate the average rate of change in height on the interval [4, 8]:**

- For the interval \([4, 8]\), the time is from 4 to 8 seconds.
- From the graph:
  - At \( t = 4 \), the height \( f(4) = 6 \) meters.
  - At \( t = 8 \), the height \( f(8) = 0 \) meters.

The average rate of change is:

\[
\frac{f(8) - f(4)}{8 - 4} = \frac{0 - 6}{8 - 4} = \frac{-6}{4} = -1.5 \, \text{meters per second}.
\]

### c) **Discuss the similarities and differences in your answers to parts a) and b):**

- **Similarity:** 
  - The **magnitude** of the average rate of change in both intervals is the same: \( 1.5 \, \text{meters per second} \). This indicates that the speed of the Ferris wheel's motion is the same in both intervals.

- **Difference:**
  - The **direction** of the rate of change is different. 
    - In part (a), the rate is positive \(+1.5\), meaning the person is ascending (height is increasing).
    - In part (b), the rate is negative \(-1.5\), meaning the person is descending (height is decreasing).

Would you like further clarification or additional details on any part of this solution?

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Here are five related questions to explore further:

1. What is the average rate of change over the interval \([0, 8]\)?
2. How would the rate of change differ if we considered a larger time interval, such as \([8, 16]\)?
3. Can you describe what the rate of change would be at the peak of the graph (e.g., at \( t = 6 \))?
4. How does the periodic nature of the graph affect the rates of change over different intervals?
5. What would the instantaneous rate of change be at \( t = 4 \) or \( t = 8 \)?

**Tip:** The average rate of change gives you a sense of the "overall trend" between two points, while the instantaneous rate of change (like the derivative) shows how fast something is changing at a specific moment.

The graph shows the height above the ground of a person riding a Ferris wheel. a) Calculate the average rate of change in height on the interval [0, 4]. b) Calculate the average rate of change in height on the interval [4, 8]. c) Discuss the similarities and differences in your answers to parts a) and b).

Learn how to calculate the average rate of change in height on a Ferris wheel ride. This problem involves finding the average rate of change over intervals using the graph of a rider's height above ground. Suitable for students in Grades 9-12.

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