To determine which interval has the greatest average rate of change of $r$ with respect to $O$, we need to use the formula for the average rate of change:

$\text{Average Rate of Change} = \frac{r(\text{end point}) - r(\text{start point})}{\text{end point} - \text{start point}}$

Without specific numerical values or a graph showing the relationship between $r$ and $O$, it's impossible to calculate the exact rates of change for each interval. However, to identify the greatest average rate of change, you would look for the interval with the largest difference in $r$-values relative to the difference in the $O$-values.

To proceed with this problem:

1. If you have a graph or any specific data for $r$ and $O$, you can apply this formula to each of the intervals.
2. Compare the values you get from each interval and identify which one is the greatest.

Do you have the graph or specific values to analyze, or would you like more guidance on how to approach this kind of problem with example data?

Follow-up questions:

1. Do you have specific data or a graph for $r$ and $O$?
2. How do you plot the data points of $r$ vs $O$?
3. Can you find the differences in $r$-values and $O$-values for each interval?
4. What does the rate of change tell you about the behavior of the function over different intervals?
5. How would the rate of change relate to the slope of the graph between two points?

Tip:

The average rate of change can be interpreted as the slope of the secant line between two points on a curve. Identifying the steepest slope between two points will give you the greatest average rate of change.