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### (D) Behavior of the Average Cost as Production Increases
As $ x \to \infty $, the term $ \frac{300}{x} $ in $ \overline{C}(x) = 240 + \frac{300}{x} $ approaches zero. Therefore, the average cost per board $ \overline{C}(x) $ approaches the value of 240 as $ x $ increases indefinitely.

### Summary of Answers
- **(A)** Cost function: $ C(x) = 240x + 300 $.
- **(B)** Average cost function: $ \overline{C}(x) = 240 + \frac{300}{x} $.
- **(C)** Graph shows a horizontal asymptote at $ y = 240 $.
- **(D)** As production $ x $ increases, the average cost per board approaches $240.

Do you need further details on any part of the solution?

### Related Questions
1. How would the graph change if the fixed cost were higher?
2. What if the cost per board wasn’t constant? How would we approach it?
3. How does increasing fixed costs affect the break-even point?
4. Can we derive a break-even formula using the cost function?
5. What if we included additional operational costs? How would that impact $ \overline{C}(x) $?

**Tip:** Always check for asymptotes when analyzing cost functions, as they reveal long-term trends in average costs.

A company manufacturing surfboards has fixed costs of $300 per day and total costs of $5,100 per day for a daily output of 20 boards. Solve parts A-D as follows:

(A) Assuming that the total cost per day C(x) is linearly related to the total output per day x, write an equation for the cost function.

(B) The average cost per board for an output of x boards is given by C̅(x) = C(x)/x. Find the average cost function.

(C) Sketch a graph of the average cost function, including any asymptotes, for 1 ≤ x ≤ 30.

(D) What does the average cost per board tend to as production increases?

This page provides a detailed solution for calculating the average cost in a surfboard manufacturing scenario, with a cost function C(x) = 240x + 300. Learn how to determine the average cost per board, graph the function, and understand the behavior as production increases. Suitable for high school students in Grades 11-12.

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