Let's address each part of the problem step-by-step.

Part (A): Cost Function

We know two key pieces of information:

• The fixed costs are $300 per day.
• The total costs are $5,100 per day when 20 boards are produced.

We assume the total cost per day $C(x)$ is linearly related to the total output per day $x$. This means the cost function $C(x)$ can be expressed as: $C(x) = mx + b$

where:

• $m$ is the variable cost per unit (the cost of producing one additional board).
• $b$ is the fixed cost.

Given:

1. Fixed costs $b = 300$.
2. When $x = 20$, $C(20) = 5100$.

Substituting $x = 20$ into the cost function: $5100 = 20m + 300$

Solving for $m$: $5100 - 300 = 20m$ $4800 = 20m$ $m = \frac{4800}{20}$ $m = 240$

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

Part (B): Average Cost Function

The average cost per board for an output of $x$ boards is given by: $\bar{C}(x) = \frac{C(x)}{x}$

Substitute $C(x) = 240x + 300$ into this formula: $\bar{C}(x) = \frac{240x + 300}{x}$ $\bar{C}(x) = 240 + \frac{300}{x}$

Part (C): Graph of the Average Cost Function

To sketch the graph of the average cost function $\bar{C}(x) = 240 + \frac{300}{x}$, we need to understand its behavior and asymptotes.

1. Vertical Asymptote: There is a vertical asymptote at $x = 0$ because $\bar{C}(x)$ is undefined at $x = 0$.

2. Horizontal Asymptote: As $x \to \infty$, $\frac{300}{x} \to 0$, so $\bar{C}(x) \to 240$. Therefore, the horizontal asymptote is $y = 240$.

The graph of $\bar{C}(x) = 240 + \frac{300}{x}$ will be a hyperbola that approaches $y = 240$ as $x$ increases and will tend to infinity as $x$ approaches zero from the right. Here is a rough sketch:

     ^ 
     | 
300+ | 
     | 
     | 
     | 
     |                  
     |                -------------------
 240 +---------------------------------------------->
     |
     |
     |
     +-------------------------------------------------> x

Summary

• Cost Function: $C(x) = 240x + 300$
• Average Cost Function: $\bar{C}(x) = 240 + \frac{300}{x}$

The graph of the average cost function shows a hyperbola with a horizontal asymptote at $y = 240$ and a vertical asymptote at $x = 0$.

Would you like further details or have any questions on this? Here are some related questions you might find interesting:

1. How would the cost function change if the fixed costs increased to $500?
2. What would be the new average cost function if the variable cost per board changed to $260?
3. How does the graph of the average cost function change if the total output increases to 30 boards?
4. What happens to the average cost function if the fixed costs are reduced by 50%?
5. Can the cost function be non-linear in real-world scenarios? If so, how?

Tip: When working with cost functions, understanding the distinction between fixed and variable costs is crucial for accurately modeling and predicting expenses.