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### Problem Overview
Madelyn's total cost $ C $ to manufacture phones includes:
- **Fixed costs** (e.g., rent for the building and equipment), and
- **Variable costs** (the cost to produce each phone).

The graph shows $ C $, the total cost (in dollars), as a function of $ p $, the number of phones produced.

### Objective
- Write an equation for $ C $ in terms of $ p $ based on the graph.
- Determine the slope of the graph and interpret it.

### Key Insights from the Graph
1. **Y-Intercept**: The total cost $ C $ when $ p = 0 $ phones are manufactured. From the graph, this value is $ C = 600 $. This represents the fixed cost, as no phones are made yet.
   
2. **Slope of the Graph**: We can calculate the slope by using two points from the line. From the graph, we can pick two points:
   - Point 1: $ (0, 600) $
   - Point 2: $ (10, 2600) $

   The slope $ m $ is calculated as:
   \[
   m = \frac{C_2 - C_1}{p_2 - p_1} = \frac{2600 - 600}{10 - 0} = \frac{2000}{10} = 200
   \]
   This means the slope is 200, which represents the cost per phone manufactured.

### Equation of the Line
The equation of a line is given by:
\[
C = mp + b
\]
where:
- $ m $ is the slope (200),
- $ b $ is the y-intercept (600).

Thus, the equation becomes:
\[
C = 200p + 600
\]

### Interpretation of the Slope
The slope of the graph, 200, represents the **variable cost** per phone. This means that for every phone Madelyn produces, the total cost increases by $200.

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Do you need more details or have any questions?

Here are 5 related questions to further explore this topic:
1. How would the equation change if the fixed costs increased to $800?
2. What does it mean if the slope of the line decreases to 150?
3. How would the graph change if Madelyn gets a discount on production costs?
4. What is the total cost if 15 phones are produced?
5. How would the graph look if no fixed costs were involved?

**Tip:** Always pay attention to the slope in cost-related problems, as it often represents the rate of change or variable cost per unit.

Madelyn is in the business of manufacturing phones. She must pay a daily fixed cost to rent the building and equipment, and also pays a cost per phone produced for materials and labor. Let C represent the total cost, in dollars, of producing p phones in a given day. A graph of C is shown. Write an equation for C and state the slope of the graph and its interpretation.

Learn how to write a linear equation for the total cost of manufacturing phones, including fixed costs and variable costs per unit. This problem explains the slope-intercept form, with a detailed step-by-step solution and interpretation of the slope in the context of business costs. Suitable for Grades 9-11.

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