https://r2.erweima.ai/mathbot/image/1727106254155r4k1y1qs.jpg

The image provides a graph showing two lines representing the cost total \( C \) and the utility total \( U \) for a certain company. The graph's axes represent the number of units (\( x \)) on the horizontal axis and the corresponding values of \( C \) and \( U \) in soles on the vertical axis. The problem asks to determine whether three given mathematical expressions are true or false based on the graph.

### Let's analyze each statement one by one:

#### a) \( C = 0.5x + 20 \)
- This is supposed to represent the total cost \( C \).
- The line \( C \) has a slope of \( 0.5 \) (since it rises 10 units over 20 units horizontally) and a \( y \)-intercept of \( 20 \). Therefore, the equation matches the line on the graph.
- **Conclusion**: This statement is **true**.

#### b) \( U = \frac{4}{3}x - 20 \)
- This is supposed to represent the total utility \( U \).
- The line \( U \) has a slope of \( \frac{4}{3} \) (since it rises 20 units over 15 units horizontally) and a \( y \)-intercept of \( -20 \), which aligns with the graph.
- **Conclusion**: This statement is **true**.

#### c) "600 soles es el ingreso total por 300 unidades vendidas."
- We need to calculate the total income for 300 units using the cost function or utility function.
- Substituting \( x = 300 \) into the utility function \( U = \frac{4}{3}x - 20 \):
  \[
  U(300) = \frac{4}{3} \times 300 - 20 = 400 - 20 = 380 \, \text{soles}
  \]
- This does not equal 600 soles, so the statement is incorrect.
- **Conclusion**: This statement is **false**.

### Final Results:
- a) **True**
- b) **True**
- c) **False**

Would you like more details or have any questions about the solution?

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Here are 5 related questions:
1. How do you calculate the slope of a line from a graph?
2. What is the interpretation of the \( y \)-intercept in cost functions?
3. How can we use linear equations to predict future costs and revenues?
4. What is the difference between cost and utility in business economics?
5. How can graphical methods assist in verifying mathematical expressions?

**Tip**: Always verify the units of the slope when analyzing linear functions to ensure correct interpretations!

The graph shows two lines representing total cost (C) and total utility (U) for a company. The task is to verify the correctness of the following statements:

a) C = 0.5x + 20 is the equation for total cost.
b) U = (4/3)x - 20 is the equation for total utility.
c) 600 soles is the total revenue for 300 units sold.

Justify if each statement is true or false.

Learn how to verify linear cost and utility equations from a graph. This problem involves calculating slope, y-intercepts, and understanding the relationship between units sold and revenue. Perfect for students at the high school level, exploring linear functions.

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