To determine the correct statement, let’s analyze the given information about the cost functions and the graph provided.
### Hiroto's Plan Cost Function:
- Monthly fee: $20
- Cost per text: $0.05
The cost function for Hiroto’s plan can be written as:
\[ \text{Cost}_{\text{Hiroto}} = 20 + 0.05x \]
where \( x \) represents the number of texts sent.
### Emilia's Plan Cost Function:
- Monthly fee: $10
- Cost per text: $0.25
The cost function for Emilia’s plan can be written as:
\[ \text{Cost}_{\text{Emilia}} = 10 + 0.25x \]
### Analyze the Intersection Point:
According to the graph, both plans intersect at point (50, 22.5).
1. **Both plans cost the same when 50 texts are sent.**
Substitute \( x = 50 \) into both cost equations:
\[ \text{Cost}_{\text{Hiroto}} = 20 + 0.05 \times 50 = 20 + 2.5 = 22.5 \]
\[ \text{Cost}_{\text{Emilia}} = 10 + 0.25 \times 50 = 10 + 12.5 = 22.5 \]
Both costs are $22.5 at 50 texts, which matches the intersection point given on the graph.
2. **Hiroto’s plan costs more than Emilia’s plan when more than 50 texts are sent.**
Since after 50 texts, the cost of Hermione's plan increases at a rate of $0.25 per text, while Hiroto’s plan increases at $0.05 per text, Emilia’s plan should become more expensive beyond this point.
### Evaluation of the Remaining Statements:
- **Both plans cost the same when 22 texts are sent:**
We need to check if the costs are the same at 22 texts.
\[ \text{Cost}_{\text{Hiroto}} = 20 + 0.05 \times 22 = 20 + 1.1 = 21.1 \]
\[ \text{Cost}_{\text{Emilia}} = 10 + 0.25 \times 22 = 10 + 5.5 = 15.5 \]
The costs are not the same at 22 texts.
- **Emilia’s plan costs more than Hiroto’s plan when more than 22 texts are sent:**
Emilia’s plan starts cheaper but eventually becomes more expensive as more texts are sent.
Conclusion: The true statement, based on the graph and the analysis, is:
**Both plans cost the same when 50 texts are sent.**
A graph shows the number of texts, numbered 10 to 100, on the x-axis, and the total cost in dollars, numbered 3 to 27, on the y-axis. A straight red line with a positive slope, labeled Emilia, begins at (0, 10), and a straight blue line with a positive slope, labeled Hiroto, begins at (0, 20). Both lines intersect at point (50, 22.5).
Hiroto’s texting plan costs $20 per month, plus $0.05 per text message that is sent or received. Emilia’s plan costs $10 per month and $0.25 per text. Using the graph below, which statement is true?
Hiroto’s plan costs more than Emilia’s plan when more than 50 texts are sent.
Both plans cost the same when 22 texts are sent.
Emilia’s plan costs more than Hiroto’s plan when more than 22 texts are sent.
Both plans cost the same when 50 texts are sent.
1 answer
1 answer