MANUFACTURING COST A manufacturer's total cost consists of a fixed overhead of $5,000 plus production costs of $60 per unit.

a. Express the total cost as a function of the number of units produced, and sketch its graph.

### Solution:

**a. Total Cost Function:**

Let the number of units produced be denoted by $ x $. The total cost $ C(x) $ consists of two parts:

1. **Fixed overhead cost** of $5,000 (which does not depend on the number of units produced).
2. **Variable cost per unit** of $60 (which depends on the number of units produced).

The total cost function can be written as:

\[
C(x) = 5000 + 60x
\]

where:
- $ 5000 $ is the fixed overhead cost,
- $ 60x $ is the variable production cost for $ x $ units.

**Graph of the Total Cost Function:**

The graph of $ C(x) = 5000 + 60x $ is a straight line, where:
- The y-intercept is $5000 (when $ x = 0 $, the total cost is $5000),
- The slope of the line is 60, which represents the cost increase of $60 for each additional unit produced.

### Key Points:
1. $ C(0) = 5000 $ (fixed cost)
2. $ C(10) = 5000 + 60(10) = 5600 $
3. $ C(100) = 5000 + 60(100) = 11000 $

This represents a linear relationship between the total cost and the number of units produced, with the cost increasing at a constant rate of $60 per unit.

Would you like me to create the graph of this function as well?

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### Follow-Up Questions:
1. What is the total cost when 50 units are produced?
2. At what production level does the total cost reach $10,000?
3. How does the fixed overhead affect the total cost function?
4. What is the marginal cost of producing one additional unit?
5. What would happen to the total cost if the fixed overhead increased by 10%?

**Tip:** For linear cost functions, the slope represents the marginal cost, which is the cost added for each extra unit produced.

Learn how to express the total manufacturing cost as a linear function of the number of units produced. This problem involves calculating the fixed overhead of $5,000 and variable production costs of $60 per unit. Explore key algebraic concepts with detailed step-by-step solutions and a graph representation. Suitable for Grades 8-10.

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