The problem displayed in the image involves finding the average rate of change for a given cost function in dollars, which is denoted as $C(x)$, over a specified interval. It says:

"The cost in dollars to produce televisions in a factory is given by the function $C(x) = 50x + 100$. Determine the average rate of change of the cost function, from $x = 10$ to $x = 4$."

To solve the average rate of change:

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 case:

• The function is $C(x) = 50x + 100$
• The interval is from $x = 10$ to $x = 4$

Let's calculate the average rate of change.

1. Find $C(10)$: $C(10) = 50(10) + 100 = 500 + 100 = 600$

2. Find $C(4)$: $C(4) = 50(4) + 100 = 200 + 100 = 300$

3. Compute the average rate of change: $\text{Average rate of change} = \frac{C(4) - C(10)}{4 - 10} = \frac{300 - 600}{4 - 10} = \frac{-300}{-6} = 50$

Thus, the average rate of change is 50.

Would you like a further explanation or have any questions?

Here are 5 related questions for further exploration:

1. How would the average rate of change differ if the interval was from $x = 8$ to $x = 2$?
2. What is the significance of the average rate of change in this context?
3. How does the slope of a linear function relate to its rate of change?
4. Can you generalize the formula for the average rate of change for any linear function?
5. How would the average rate of change change if the function was quadratic instead of linear?

Tip: The average rate of change for a linear function is always constant, as it's equal to the slope of the line.