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We are given the cost function \( C(x) = 1000 + 5.7x + 0.7x^2 \), and we need to calculate the average rate of change of \( C \) with respect to \( x \) as the production level increases from \( x = 100 \) to \( x = 120 \).

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

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

In this case, \( a = 100 \) and \( b = 120 \).

### Step 1: Calculate \( C(100) \)

\[
C(100) = 1000 + 5.7(100) + 0.7(100)^2
\]

### Step 2: Calculate \( C(120) \)

\[
C(120) = 1000 + 5.7(120) + 0.7(120)^2
\]

### Step 3: Find the average rate of change

\[
\frac{C(120) - C(100)}{120 - 100}
\]

I'll compute these values now.The cost at \( x = 100 \) is \( C(100) = 8570 \), and at \( x = 120 \), \( C(120) = 11764 \).

The average rate of change of \( C \) from \( x = 100 \) to \( x = 120 \) is:

\[
\frac{C(120) - C(100)}{120 - 100} = \frac{11764 - 8570}{120 - 100} = 159.7
\]

Thus, the correct answer is **D. 159.7**.

Would you like a further explanation or have any additional questions?

Here are 5 related questions you might explore:
1. How is the derivative related to the average rate of change?
2. What is the instantaneous rate of change of the cost function at \( x = 100 \)?
3. How would the average rate of change differ if the interval was \( x = 120 \) to \( x = 140 \)?
4. Can you explain the significance of the quadratic term \( 0.7x^2 \) in the cost function?
5. How does the constant 1000 in the cost function affect the average rate of change?

**Tip**: The average rate of change is essentially the slope of the secant line between two points on the graph of the function.

The cost of producing x units of a certain commodity is C(x) = 1000 + 5.70x + 0.7x^2. What is the average rate of change of C with respect to x when the production level is raised from x = 100 to x = 120?

Calculate the average rate of change for the cost function C(x) = 1000 + 5.70x + 0.7x^2 over the interval from x = 100 to x = 120. This problem involves the rate of change formula and quadratic equations, suitable for students in Grades 9-12.

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