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To find the average rate of change for the function \( f(x) = x^2 + 10x \) between \( x = 0 \) and \( x = 9 \), we can use the formula provided in option C:

\[
\text{Average rate of change} = \frac{f(9) - f(0)}{9 - 0}
\]

### Step-by-Step Solution:

1. **Calculate \( f(9) \):**
   \[
   f(9) = 9^2 + 10 \cdot 9 = 81 + 90 = 171
   \]

2. **Calculate \( f(0) \):**
   \[
   f(0) = 0^2 + 10 \cdot 0 = 0
   \]

3. **Apply the formula:**
   \[
   \text{Average rate of change} = \frac{f(9) - f(0)}{9 - 0} = \frac{171 - 0}{9} = \frac{171}{9} = 19
   \]

### Answer:
The average rate of change is \( 19 \).

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

### Related Questions:
1. How does the average rate of change differ from the instantaneous rate of change?
2. What is the geometric interpretation of the average rate of change on a graph?
3. Can the average rate of change be used to estimate values outside the interval \( [0, 9] \)?
4. How would the answer change if the interval were from \( x = 3 \) to \( x = 9 \)?
5. How does this calculation apply to other quadratic functions?

### Tip:
The average rate of change over an interval for any function is the slope of the secant line between the two points on the graph of the function.

Find the average rate of change for the given function, f(x) = x^2 + 10x, between x = 0 and x = 9.

Learn how to find the average rate of change for the function f(x) = x^2 + 10x between x = 0 and x = 9. This problem demonstrates the use of the average rate of change formula with a step-by-step solution, suitable for high school students.

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