Two cell phone companies offer different monthly plans. Company A charges a $30 monthly fee plus $0.10 per minute of...

1. TRANSLATE the intersection point information

• Given information:
  • Company A: $30 monthly fee + $0.10 per minute
  • Company B: $50 monthly fee + $0.05 per minute
  • Intersection point: (400, 70)

• What this tells us: At \(\mathrm{x = 400}\) minutes, both cost functions equal \(\mathrm{y = \$70}\)

2. INFER what intersection means mathematically

• When two graphs intersect, the functions have the same output value at that input
• This means both companies charge the same amount at 400 minutes of usage
• This is the "break-even" point where neither plan is cheaper

3. TRANSLATE this back to the real-world context

• The point (400, 70) means: "When someone uses exactly 400 minutes, both Company A and Company B will charge exactly $70"

Let me verify this makes sense:
Company A: \(\mathrm{\$30 + \$0.10(400) = \$30 + \$40 = \$70}\) ✓
Company B: \(\mathrm{\$50 + \$0.05(400) = \$50 + \$20 = \$70}\) ✓

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse what the coordinates represent or misread the intersection point's meaning.

Some students see (400, 70) and think "400 represents cost and 70 represents minutes" - completely backwards. Others might think the intersection shows a $400 difference between companies or that it represents Company A costing $70 less than Company B. These fundamental misreadings of coordinate meaning make it impossible to select the correct interpretation.

This may lead them to select Choice C (At 400 minutes, Company A costs $70 less than Company B) or Choice D (The difference in monthly fees between the companies is $400).

Second Most Common Error:

Poor INFER reasoning about intersection meaning: Students don't connect that intersection points represent equality between functions.

Students might correctly read the coordinates but fail to understand that intersection means "both functions have the same value." Instead, they might think the intersection shows when one company becomes more expensive than the other, leading to confusion about timing relationships.

This may lead them to select Choice A (After 400 minutes, Company A becomes more expensive than Company B).

The Bottom Line:

This problem requires students to bridge pure mathematical concepts (intersection points) with real-world business contexts (cost comparisons). Success depends on accurately translating coordinates into meaningful business language while understanding that intersections represent equality, not differences.