We are tasked with finding the linear cost function $C(x)$, given the following information:

• Fixed cost = $200.
• Cost of producing 80 items = $3,400.

A linear cost function has the form:

$C(x) = mx + b$

Where:

• $m$ is the variable cost per item (slope),
• $b$ is the fixed cost,
• $x$ is the number of items produced.

Step 1: Identify the fixed cost

From the problem, we know the fixed cost is $200. Thus, $b = 200$.

Step 2: Use the information to find the variable cost per item

We are given that producing 80 items costs $3,400. Therefore, we can plug this into the linear equation:

$C(80) = 80m + 200 = 3400$

Now, solve for $m$:

$80m + 200 = 3400$

$80m = 3400 - 200 = 3200$

$m = \frac{3200}{80} = 40$

Step 3: Write the final cost function

The variable cost per item is $40, and the fixed cost is $200. Thus, the cost function is:

$C(x) = 40x + 200$

Would you like further clarification or details on any part of this? Here are 5 related questions for deeper understanding:

1. How does a change in the fixed cost affect the overall cost function?
2. What is the cost for producing 100 items using the function $C(x) = 40x + 200$?
3. How would the function change if the cost for 80 items was different?
4. Can the linear cost function apply if the variable cost changes with the number of items produced?
5. How would you graph the linear cost function?

Tip: The slope of a linear function represents the rate of change, which in cost functions is the variable cost per item.