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To solve the problem, we need to calculate the average rate of change of the function \( f(x) = 6 \sin(x) \) over the interval \(\left[\frac{\pi}{6}, \frac{\pi}{2}\right]\).

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

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

### Step 1: Calculate \( f(b) \) and \( f(a) \)
- \( b = \frac{\pi}{2} \)
- \( a = \frac{\pi}{6} \)

We are given \( f(x) = 6 \sin(x) \).

\[
f(b) = f\left(\frac{\pi}{2}\right) = 6 \sin\left(\frac{\pi}{2}\right) = 6(1) = 6
\]

\[
f(a) = f\left(\frac{\pi}{6}\right) = 6 \sin\left(\frac{\pi}{6}\right) = 6\left(\frac{1}{2}\right) = 3
\]

### Step 2: Calculate the difference \( f(b) - f(a) \)

\[
f(b) - f(a) = 6 - 3 = 3
\]

### Step 3: Calculate \( b - a \)

\[
b - a = \frac{\pi}{2} - \frac{\pi}{6} = \frac{3\pi}{6} - \frac{\pi}{6} = \frac{2\pi}{6} = \frac{\pi}{3}
\]

### Step 4: Compute the average rate of change

\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} = \frac{3}{\frac{\pi}{3}} = 3 \times \frac{3}{\pi} = \frac{9}{\pi}
\]

Thus, the average rate of change of \( f(x) = 6 \sin(x) \) over the interval \(\left[\frac{\pi}{6}, \frac{\pi}{2}\right]\) is:

\[
\boxed{\frac{9}{\pi}}
\]

Would you like further explanation or details?

Here are 5 related questions for practice:
1. What is the average rate of change of \( f(x) = 4\sin(x) \) on the interval \( \left[\frac{\pi}{4}, \frac{\pi}{2}\right] \)?
2. How does the average rate of change of \( \sin(x) \) over \([0, \pi]\) compare to that over \( [\pi, 2\pi] \)?
3. Can you generalize the formula for the average rate of change of \( f(x) = A\sin(x) \) over any interval \([a, b]\)?
4. What is the physical interpretation of the average rate of change for a sine wave over an interval?
5. How would the solution change if the interval were \([0, \frac{\pi}{3}]\)?

**Tip**: When calculating average rate of change, always simplify the difference in the function values and the interval first before dividing to make the process smoother.

Given the function f(x) = 6sin(x), find the average rate of change of f(x) on the interval [π/6, π/2]. Your final answer must be simplified, which means no fractions in your fraction.

Calculate the average rate of change for the function f(x) = 6sin(x) over the interval [π/6, π/2]. This problem involves key trigonometric concepts such as the sine function and the average rate of change formula. A step-by-step solution is provided for high school level math.

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