A company manufacturing surfboard has fixed costs of $300 per day and total costs of $5,100 per
day at a daily output of 20 boards.
(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.

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.

A company manufacturing surfboard has fixed costs of $300 per day and total costs of $5,100 per day at a daily output of 20 boards. (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.

Explore how to calculate the linear cost function for a surfboard manufacturing company with fixed costs of $300 and total costs of $5,100 at a daily output of 20 boards. Learn how to derive the equation C(x) = 240x + 300, representing the total cost per day, and understand key algebraic concepts related to cost functions. Suitable for students in Grades 9-11.

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