At a trailhead, the air temperature is 68 degrees Fahrenheit. As a hiker ascends, the temperature decreases by 3.6 degrees...

1. TRANSLATE the problem information

• Given information:
  • Starting temperature at trailhead: \(68°\mathrm{F}\)
  • Temperature decreases by \(3.6°\mathrm{F}\) for every \(1,000\) feet of elevation gain
  • Lookout point elevation: \(4,500\) feet above trailhead
• What we need to find: Temperature at the lookout point

2. INFER the approach

• This is a proportional relationship problem - temperature decreases at a constant rate with elevation
• Strategy: Find how many 1,000-foot intervals we're climbing, then calculate total temperature decrease
• Finally, subtract that decrease from the starting temperature

3. Calculate elevation intervals

• Number of 1,000-foot intervals = \(4,500 \div 1,000 = 4.5\) intervals

4. SIMPLIFY to find total temperature decrease

• Total decrease = \(4.5 \times 3.6°\mathrm{F} = 16.2°\mathrm{F}\) (use calculator)

5. SIMPLIFY to find final temperature

• Final temperature = \(68°\mathrm{F} - 16.2°\mathrm{F} = 51.8°\mathrm{F}\)

Answer: (A) 51.8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misinterpret the rate and think \(3.6°\mathrm{F}\) decrease applies to the entire \(4,500\) feet, rather than per \(1,000\) feet.

They might calculate: \(68 - 3.6 = 64.4°\mathrm{F}\)

This may lead them to select Choice (C) (64.4)

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the problem but make arithmetic errors in the multiplication \(4.5 \times 3.6\), perhaps calculating \(4.5 \times 3.6 = 14.4\) instead of \(16.2\).

This would give them: \(68 - 14.4 = 53.6°\mathrm{F}\)

This may lead them to select Choice (B) (53.6)

The Bottom Line:

This problem requires careful attention to the rate structure ("per 1,000 feet") and precise arithmetic execution. The key insight is recognizing that the given rate must be scaled up proportionally for the actual elevation gain.