Question
Sam has $50.00 to spend this month on his cell phone plan. Calls cost $0.10 per minute, and text messages cost $0.05 per message.
Which of the following graphs represents this situation?
J.
K.
L.
M.
Which of the following graphs represents this situation?
J.
K.
L.
M.
Answers
GPT-4o mini
To determine which graph represents Sam's spending situation on his cell phone plan, we first need to define the relationship between call minutes and text messages with respect to his budget.
1. **Define Variables**: Let:
- \( x \) = number of minutes for calls
- \( y \) = number of text messages
2. **Cost Equation**: Sam's total spending must not exceed $50. The cost for calls is $0.10 per minute, and the cost for text messages is $0.05 per message. Therefore, we can write the equation:
\[
0.10x + 0.05y \leq 50
\]
3. **Rearranging the Equation**: To understand the relationship between \( x \) and \( y \), we can express \( y \) in terms of \( x \):
\[
0.05y \leq 50 - 0.10x
\]
\[
y \leq \frac{50 - 0.10x}{0.05} = 1000 - 2x
\]
This shows that \( y \) (the number of text messages) decreases as \( x \) (the number of call minutes) increases.
4. **Graph Characteristics**:
- The line \( y = 1000 - 2x \) is a downward sloping line (indicating that as call minutes increase, text messages can decrease).
- The intercepts are where \( x \) or \( y \) equals zero:
- When \( x = 0 \): \( y = 1000 \)
- When \( y = 0 \): \( x = 500 \)
To determine which graph corresponds to this linear equation and characteristic, you need to identify a graph that:
- Has a negative slope.
- Crosses the y-axis at 1000.
- Crosses the x-axis at 500.
Please provide the graphs (J, K, L, M), so I can help you identify which one represents the situation described above.
1. **Define Variables**: Let:
- \( x \) = number of minutes for calls
- \( y \) = number of text messages
2. **Cost Equation**: Sam's total spending must not exceed $50. The cost for calls is $0.10 per minute, and the cost for text messages is $0.05 per message. Therefore, we can write the equation:
\[
0.10x + 0.05y \leq 50
\]
3. **Rearranging the Equation**: To understand the relationship between \( x \) and \( y \), we can express \( y \) in terms of \( x \):
\[
0.05y \leq 50 - 0.10x
\]
\[
y \leq \frac{50 - 0.10x}{0.05} = 1000 - 2x
\]
This shows that \( y \) (the number of text messages) decreases as \( x \) (the number of call minutes) increases.
4. **Graph Characteristics**:
- The line \( y = 1000 - 2x \) is a downward sloping line (indicating that as call minutes increase, text messages can decrease).
- The intercepts are where \( x \) or \( y \) equals zero:
- When \( x = 0 \): \( y = 1000 \)
- When \( y = 0 \): \( x = 500 \)
To determine which graph corresponds to this linear equation and characteristic, you need to identify a graph that:
- Has a negative slope.
- Crosses the y-axis at 1000.
- Crosses the x-axis at 500.
Please provide the graphs (J, K, L, M), so I can help you identify which one represents the situation described above.