The average rate of change of a function $f(x)$ over an interval $[a, b]$ is calculated using the formula:

$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$

Given the function $f(x) = 3x + 10$ and the interval $[5, 10]$:

1. Calculate $f(5)$: $f(5) = 3(5) + 10 = 15 + 10 = 25$

2. Calculate $f(10)$: $f(10) = 3(10) + 10 = 30 + 10 = 40$

3. Apply the average rate of change formula: $\text{Average Rate of Change} = \frac{f(10) - f(5)}{10 - 5} = \frac{40 - 25}{10 - 5} = \frac{15}{5} = 3$

So, the average rate of change of the function $f(x) = 3x + 10$ from $x = 5$ to $x = 10$ is $\boxed{3}$.

Would you like more details or have any questions?

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Here are some related questions:

1. How does the average rate of change relate to the derivative of a function?
2. What is the geometric interpretation of the average rate of change for a linear function?
3. How would the average rate of change differ for a quadratic function?
4. Can the average rate of change be negative? If so, under what conditions?
5. How do you find the average rate of change for a non-linear function?
6. What is the average rate of change for $f(x) = x^2$ over the interval $[2, 5]$?
7. What is the significance of the interval $[a, b]$ when calculating the average rate of change?
8. How does the average rate of change differ from the instantaneous rate of change?

Tip: When dealing with linear functions, the average rate of change over any interval will always equal the slope of the line.