A chase between two cyclists is modeled by the function \(\mathrm{s(t) = -2t^2 + 20t + 50}\), where \(\mathrm{s(t)}\) is...

1. TRANSLATE the mathematical terminology to real-world meaning

• Given information:
  • \(\mathrm{s(t) = -2t^2 + 20t + 50}\) represents separation distance (meters) between cyclists
  • t represents time (seconds) after chase begins
  • Question asks about "positive x-intercept"

• TRANSLATE: An x-intercept occurs when \(\mathrm{y = 0}\), so when \(\mathrm{s(t) = 0}\)

2. INFER what zero separation means in real-world context

• If \(\mathrm{s(t) = 0}\), then separation distance = 0 meters
• Zero separation means the cyclists are at exactly the same position
• In other words: the cyclists are side by side

3. APPLY CONSTRAINTS to select the meaningful intercept

• The positive x-intercept represents a time after the chase began (\(\mathrm{t \gt 0}\))
• The negative x-intercept would represent a time before the chase started (not physically meaningful)

Answer: D - The time at which the cyclists are side by side

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse different features of quadratic graphs. They might think about:

• y-intercept (what happens when t = 0) → leads to selecting Choice A (The initial separation)
• Vertex of parabola (maximum/minimum point) → leads to selecting Choice B or C

Second Most Common Error:

Inadequate INFER reasoning: Students correctly identify that x-intercept means \(\mathrm{s(t) = 0}\), but fail to connect this to the real-world meaning. They might think "zero" relates to when one cyclist stops moving rather than when they meet.

This may lead them to select Choice E (The time at which Cyclist A stops)

The Bottom Line:

This problem tests whether students can bridge mathematical graph features with real-world context. The key insight is recognizing that "separation distance equals zero" means "same position" rather than "stopped motion."