Let's work through each part step by step:

Part A: Cost Function

The cost function, $C(x)$, is the total cost of producing $x$ mountain bikes.

We are told:

• The fixed monthly cost is $400,000 (regardless of how many bikes are produced).
• The variable cost is $100 per bike.

The total cost function is: $C(x) = 400{,}000 + 100x$ where $x$ is the number of mountain bikes produced.

Part B: Average Cost Function

The average cost function is the cost per bike when producing $x$ bikes. This is the total cost divided by the number of bikes produced, $x$.

The average cost function, $\overline{C}(x)$, is: $\overline{C}(x) = \frac{C(x)}{x} = \frac{400{,}000 + 100x}{x}$

Simplifying this expression: $\overline{C}(x) = \frac{400{,}000}{x} + 100$

Part C: Find $x$ when the average cost is $300

We are asked to find how many mountain bikes must be produced each month to have an average cost of $300 per bike.

Set the average cost function equal to 300: $\overline{C}(x) = 300$ Substitute the expression for $\overline{C}(x)$: $\frac{400{,}000}{x} + 100 = 300$ Solve for $x$: $\frac{400{,}000}{x} = 300 - 100$ $\frac{400{,}000}{x} = 200$ Multiply both sides by $x$: $400{,}000 = 200x$ Solve for $x$: $x = \frac{400{,}000}{200} = 2{,}000$

Final Answer:

The company must produce 2,000 mountain bikes each month for the average cost to be $300 per bike.

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Would you like more details or have any questions?

Here are 5 related questions for deeper understanding:

1. How would the average cost change if the fixed cost increases?
2. What is the break-even point for the company if the selling price is $500 per bike?
3. How does the cost function change if the variable cost per bike rises?
4. How would the production output change if the target average cost is $250?
5. What is the total cost when 5,000 bikes are produced?

Tip: Understanding how the fixed and variable costs influence total and average cost helps in making production decisions!