A running track consists of two straight sections of length l meters and two semicircular turns with the same diameter...

1. TRANSLATE the problem setup

• Given information:
  • General track formula: \(\mathrm{T(l) = 2l + \pi d}\)
  • Specific track formula: \(\mathrm{T(l) = 2l + 100\pi}\)
  • Need to find: diameter d

• These represent the same track function with different notation

2. INFER the solution approach

• Since both expressions equal T(l), they must be identical
• This means their corresponding terms must be equal
• The straight sections: \(\mathrm{2l = 2l}\) ✓
• The curved sections: \(\mathrm{\pi d = 100\pi}\)

3. SIMPLIFY to solve for d

• From \(\mathrm{\pi d = 100\pi}\)
• Divide both sides by π: \(\mathrm{d = 100}\)

Answer: C (100)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing this as a coefficient matching problem

Students may overthink the track geometry or try to calculate the total perimeter from scratch instead of simply comparing the two given expressions. They might attempt complex calculations involving the relationship between straight and curved sections, missing that the problem directly gives them two forms of the same function to compare.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Conceptual confusion about track geometry: Misunderstanding what the curved sections represent

Students might think they need the full circumference \(\mathrm{2\pi d}\) for two semicircular turns, not recognizing that two semicircles form one complete circle with circumference \(\mathrm{\pi d}\). This could lead them to set up incorrect equations or second-guess the given formula.

This may lead them to select Choice D (200) if they incorrectly think \(\mathrm{\pi d}\) should be \(\mathrm{2\pi d}\).

The Bottom Line:

This problem tests whether students can recognize that when two expressions represent the same quantity, they can match coefficients directly rather than working from first principles.