a cost function given by 
C(x) = 20 + 50 square root x
 dollars, where x is the number of students enrolled. Find the average rate of change when the number of students enrolled is between
x = 940
 and 
x = 1,080
 with the correct units. (Round your answer to two decimal places.)

The average rate of change of a function \( C(x) \) over an interval \([x_1, x_2]\) is given by the formula:

\[
\text{Average rate of change} = \frac{C(x_2) - C(x_1)}{x_2 - x_1}
\]

In this problem, the cost function is \( C(x) = 20 + 50\sqrt{x} \), and the interval of students enrolled is from \( x_1 = 940 \) to \( x_2 = 1080 \).

### Step 1: Calculate \( C(x_1) \) and \( C(x_2) \)
First, we calculate \( C(940) \) and \( C(1080) \).

\[
C(940) = 20 + 50\sqrt{940}
\]
\[
C(1080) = 20 + 50\sqrt{1080}
\]

### Step 2: Compute the average rate of change
Substitute these values into the formula for the average rate of change:

\[
\text{Average rate of change} = \frac{C(1080) - C(940)}{1080 - 940}
\]

Let me compute these values for you.### Step 1: Calculate \( C(940) \) and \( C(1080) \)
\[
C(940) = 1552.97 \, \text{dollars}
\]
\[
C(1080) = 1663.17 \, \text{dollars}
\]

### Step 2: Compute the average rate of change
The average rate of change is:
\[
\text{Average rate of change} = \frac{1663.17 - 1552.97}{1080 - 940} = 0.79 \, \text{dollars per student} 
\]

So, the average rate of change of the cost with respect to the number of students enrolled is approximately **0.79 dollars per student**.

Would you like further details on this, or do you have any questions?

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Here are 5 related questions:
1. How do you interpret the average rate of change in this context?
2. What is the instantaneous rate of change of the cost function at a specific enrollment value?
3. How would the cost change if the number of students increased by 100?
4. How does the square root in the cost function influence the rate of change?
5. How would you model a similar cost function with a different fixed fee?

**Tip:** The average rate of change is similar to the slope of a secant line connecting two points on a curve!

A cost function is given by C(x) = 20 + 50√x dollars, where x is the number of students enrolled. Find the average rate of change when the number of students enrolled is between x = 940 and x = 1,080 with the correct units. (Round your answer to two decimal places.)

Explore how to calculate the average rate of change of a cost function, C(x) = 20 + 50√x, between x = 940 and x = 1,080. Understand how changes in student enrollment impact cost, using the average rate of change formula. Suitable for high school students in Grades 10-12.

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