To solve this problem, let's go step by step:

Part (A) - Cost Function $C(x)$

The total cost $C(x)$ is a linear function of the output $x$ (number of boards produced per day). We are given:

• Fixed costs: $300 per day (This is the cost even if no boards are produced).
• Total cost for 20 boards: $5,100 per day.

We can express the cost function as: $C(x) = mx + b$ where $m$ is the variable cost per board (the slope of the line), and $b = 300$ is the fixed cost.

Using the given information that $C(20) = 5100$, we substitute into the equation: $5100 = 20m + 300$ Now, solve for $m$: $5100 - 300 = 20m$ $4800 = 20m$ $m = 240$ So, the variable cost per board is $240.

Therefore, the total cost function is: $C(x) = 240x + 300$

Part (B) - Average Cost Function $\overline{C}(x)$

The average cost per board for an output of $x$ boards is given by: $\overline{C}(x) = \frac{C(x)}{x}$ Substitute the expression for $C(x)$: $\overline{C}(x) = \frac{240x + 300}{x} = 240 + \frac{300}{x}$

Part (C) - Graph of the Average Cost Function

The function to be graphed is: $\overline{C}(x) = 240 + \frac{300}{x}$

Asymptotes:

• As $x \to \infty$, the term $\frac{300}{x} \to 0$, so $\overline{C}(x) \to 240$. Thus, there is a horizontal asymptote at $y = 240$.
• There is a vertical asymptote at $x = 0$ because division by zero is undefined.

Graph Behavior: For $1 \leq x \leq 30$, the graph starts high when $x$ is small and approaches 240 as $x$ increases.

I can generate the graph for you to visualize this, but let's proceed to the next part first.

Part (D) - Behavior of Average Cost as Production Increases

As the production of boards $x$ increases, the term $\frac{300}{x}$ becomes smaller and smaller. Therefore, the average cost per board approaches the variable cost per board of $240. Mathematically: $\lim_{x \to \infty} \overline{C}(x) = 240$ Thus, the average cost per board tends to $240 as production increases.

Would you like a detailed graph or more information on any of these parts?

Here are 5 related questions you might explore:

1. What happens to the total cost if the fixed cost increases?
2. How would the cost function change if the variable cost per board decreases?
3. What is the break-even point where the cost equals revenue?
4. How would the average cost graph change if production increased beyond 30 boards?
5. Can you find the marginal cost, and how does it relate to the average cost?

Tip: In cost analysis, identifying fixed and variable costs is key to optimizing production and pricing decisions!