Question
Grace is comparing cell phone plans. A prepaid phone plan costs $0.20 per minute and has no monthly fee. A contracted phone plan costs $50 per month and $0.02 per minute. How will the graphs of the monthly costs of the two cell phone plans compare where x represents minutes purchased in a month?
The prepaid phone plan will have a steeper line and lower y-intercept.
The contracted phone plan will have the same steepness and a higher y-intercept.
The prepaid phone plan will have a less steep line and the same y-intercept.
The contracted phone plan will have a steeper line and same y-intercept.
The prepaid phone plan will have a steeper line and lower y-intercept.
The contracted phone plan will have the same steepness and a higher y-intercept.
The prepaid phone plan will have a less steep line and the same y-intercept.
The contracted phone plan will have a steeper line and same y-intercept.
Answers
GPT-4o mini
To compare the two cell phone plans graphically, let's find the equations for the cost of both plans in terms of the number of minutes purchased, \( x \).
1. **Prepaid Phone Plan:**
- Cost = $0.20 x (where x is the number of minutes)
- Equation: \( C_{\text{prepaid}} = 0.20x \)
2. **Contracted Phone Plan:**
- Cost = $50 (monthly fee) + $0.02 per minute
- Equation: \( C_{\text{contracted}} = 50 + 0.02x \)
Now we can analyze the characteristics of each equation:
- The **slope** of the prepaid plan is \( 0.20 \).
- The **slope** of the contracted plan is \( 0.02 \).
- The **y-intercept** of the prepaid plan is \( 0 \) (since there is no monthly fee).
- The **y-intercept** of the contracted plan is \( 50 \) (due to the monthly fee).
### Comparison:
- **Steepness:** The prepaid plan has a steeper slope (0.20 vs. 0.02), which means its cost increases more rapidly with more minutes than the contracted plan.
- **Y-Intercept:** The contracted plan has a higher y-intercept ($50 vs. $0), indicating that it has a base cost because of the monthly fee.
Given this information, we can determine how the graphs will compare:
- The prepaid phone plan will have a steeper line (higher slope) and a lower y-intercept (0).
- The contracted phone plan will have a less steep line and a higher y-intercept (50).
Thus, the correct statement about their comparison would be:
**The prepaid phone plan will have a steeper line and lower y-intercept.**
1. **Prepaid Phone Plan:**
- Cost = $0.20 x (where x is the number of minutes)
- Equation: \( C_{\text{prepaid}} = 0.20x \)
2. **Contracted Phone Plan:**
- Cost = $50 (monthly fee) + $0.02 per minute
- Equation: \( C_{\text{contracted}} = 50 + 0.02x \)
Now we can analyze the characteristics of each equation:
- The **slope** of the prepaid plan is \( 0.20 \).
- The **slope** of the contracted plan is \( 0.02 \).
- The **y-intercept** of the prepaid plan is \( 0 \) (since there is no monthly fee).
- The **y-intercept** of the contracted plan is \( 50 \) (due to the monthly fee).
### Comparison:
- **Steepness:** The prepaid plan has a steeper slope (0.20 vs. 0.02), which means its cost increases more rapidly with more minutes than the contracted plan.
- **Y-Intercept:** The contracted plan has a higher y-intercept ($50 vs. $0), indicating that it has a base cost because of the monthly fee.
Given this information, we can determine how the graphs will compare:
- The prepaid phone plan will have a steeper line (higher slope) and a lower y-intercept (0).
- The contracted phone plan will have a less steep line and a higher y-intercept (50).
Thus, the correct statement about their comparison would be:
**The prepaid phone plan will have a steeper line and lower y-intercept.**
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