We are given the following information about the company's costs:

• Fixed costs: $300$ dollars per day.
• Total costs when producing 20 surfboards per day: $5100$ dollars.
• The total cost $C(x)$ is linearly related to the output $x$, where $x$ represents the number of surfboards produced.

Step 1: General form of the cost function

Since the cost function $C(x)$ is linear, it can be written in the general form: $C(x) = mx + b$ where:

• $m$ is the marginal cost (the cost to produce one more surfboard).
• $b$ is the fixed cost (which is constant, regardless of how many surfboards are produced).

Step 2: Use the given information to find $m$ and $b$

From the problem:

• We know that the fixed cost $b = 300$.
• When $x = 20$, the total cost is $C(20) = 5100$. So, we can substitute this into the cost function to solve for $m$:

$5100 = m(20) + 300$

Step 3: Solve for $m$

Simplify and solve for $m$:

$5100 = 20m + 300$ $5100 - 300 = 20m$ $4800 = 20m$ $m = \frac{4800}{20} = 240$

Step 4: Write the cost function

Now that we know $m = 240$ and $b = 300$, the cost function is: $C(x) = 240x + 300$

This is the equation for the total cost function $C(x)$, where $x$ is the number of surfboards produced per day.

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. What is the total cost of producing 30 surfboards per day?
2. What would the marginal cost per surfboard be if total costs were $6,500 for 20 boards?
3. How does the cost function change if the fixed cost increases to $400 per day?
4. What would be the break-even point if the selling price per surfboard is $500?
5. How would you determine the cost to produce zero surfboards in this model?

Tip: In cost functions, the slope (marginal cost) tells you how much each additional unit adds to total cost, and the y-intercept represents fixed costs.