Manufacturing companies usually have two types of costs. Fixed costs, sometimes called overhead, are costs that are not related to the number of items manufactured. Fixed costs include things like rent, a base level of utilities, and a base level of salaries and must be paid even if the company does not make any products. Variable costs depend on the number of items manufactured. Variable costs include things like parts and raw materials for the products, higher utilities, and higher salaries. As a manufacturing company increases the number of items it produces, the fixed costs do not change, but the variable costs increase.

One company manufactures high-end racing bicycles. The monthly fixed costs are $3530. The total cost to manufacture 23 bikes is $22275.  Assume that total cost, C, is linearly related to the number of bicycles, x, that the company manufactures.

A)  The cost function is:  
     Remember to use function notation.

B)  The total cost to manufacture 14 bikes is $

C)  The cost to produce each additional bike is $

To solve the problem, we need to establish the linear cost function and then use it to answer the given questions.

### A) The Cost Function

The total cost $ C(x) $ is composed of two parts: the fixed cost and the variable cost per bike. Since the problem states that the total cost is linearly related to the number of bikes, we can express the cost function as:
\[
C(x) = \text{fixed cost} + \text{(variable cost per bike)} \times x
\]
Where:
- $ C(x) $ is the total cost when producing $ x $ bikes.
- The fixed cost is $3530.
- We are given that the total cost to manufacture 23 bikes is $22,275, so we can use this to find the variable cost per bike.

Let the variable cost per bike be $ m $. Therefore, the total cost function is:
\[
C(x) = 3530 + mx
\]
We know that $ C(23) = 22275 $, so we can substitute $ x = 23 $ into the equation to solve for $ m $:
\[
22275 = 3530 + 23m
\]
\[
22275 - 3530 = 23m
\]
\[
18745 = 23m
\]
\[
m = \frac{18745}{23} = 815
\]
So, the variable cost per bike is $815.

Thus, the cost function is:
\[
C(x) = 3530 + 815x
\]

### B) The total cost to manufacture 14 bikes

Now, using the cost function $ C(x) = 3530 + 815x $, we substitute $ x = 14 $:
\[
C(14) = 3530 + 815 \times 14
\]
\[
C(14) = 3530 + 11410 = 14940
\]
So, the total cost to manufacture 14 bikes is **$14,940**.

### C) The cost to produce each additional bike

The cost to produce each additional bike is the **variable cost per bike**, which is the slope of the linear cost function. From part A, we found that the variable cost is **$815 per bike**.

---

### Summary:
A) The cost function is $ C(x) = 3530 + 815x $.

B) The total cost to manufacture 14 bikes is **$14,940**.

C) The cost to produce each additional bike is **$815**.

Would you like more details on any of these steps or concepts? Here are some related questions you might consider:

1. How would the cost function change if the fixed costs increased by $1000?
2. What would be the total cost to manufacture 30 bikes?
3. How many bikes can be manufactured if the budget is $40,000?
4. What happens to the cost function if the variable cost per bike decreases by $50?
5. How would the function change if the company gave discounts for bulk orders?

**Tip:** The slope in a linear function always represents the rate of change—here, it represents how much the cost increases for each additional bike.

Manufacturing companies usually have two types of costs. Fixed costs, sometimes called overhead, are costs that are not related to the number of items manufactured. Fixed costs include things like rent, a base level of utilities, and a base level of salaries and must be paid even if the company does not make any products. Variable costs depend on the number of items manufactured. Variable costs include things like parts and raw materials for the products, higher utilities, and higher salaries. As a manufacturing company increases the number of items it produces, the fixed costs do not change, but the variable costs increase. One company manufactures high-end racing bicycles. The monthly fixed costs are $3530. The total cost to manufacture 23 bikes is $22275. Assume that total cost, C, is linearly related to the number of bicycles, x, that the company manufactures. A) The cost function is: Remember to use function notation. B) The total cost to manufacture 14 bikes is $ C) The cost to produce each additional bike is $

Explore the linear cost function for a manufacturing company producing high-end bicycles. Learn how to calculate total costs based on fixed costs of $3530 and variable costs of $815 per bike. This problem is perfect for students learning about cost analysis and linear functions. Suitable for Grades 9-12.

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