A research laboratory maintains sensitive equipment at temperatures between 68°F and 77°F, inclusive, for optimal performance. A scientist monitors th...

1. TRANSLATE the problem information

• Given information:
  • Acceptable temperature range: \(68°\mathrm{F}\) to \(77°\mathrm{F}\), inclusive
  • Conversion formula: \(\mathrm{F} = \frac{9\mathrm{C}}{5} + 32\)
  • Need to find the equivalent Celsius range
• What this tells us: We need to convert both the minimum (\(68°\mathrm{F}\)) and maximum (\(77°\mathrm{F}\)) temperatures to Celsius, then express the result as an inequality.

2. INFER the approach

• Since we want the Celsius range, we need to solve the conversion equation for C at both temperature endpoints
• The word "inclusive" means we'll use ≤ and ≥ symbols, not < and >
• We'll convert each Fahrenheit value separately, then combine into one inequality

3. SIMPLIFY to find the minimum Celsius temperature

• Start with \(68°\mathrm{F}\):

\(68 = \frac{9\mathrm{C}}{5} + 32\)

\(68 - 32 = \frac{9\mathrm{C}}{5}\)

\(36 = \frac{9\mathrm{C}}{5}\)

\(36 \times 5 = 9\mathrm{C}\)

\(180 = 9\mathrm{C}\)

\(\mathrm{C} = 20°\mathrm{C}\)

4. SIMPLIFY to find the maximum Celsius temperature

• Start with \(77°\mathrm{F}\):

\(77 = \frac{9\mathrm{C}}{5} + 32\)

\(77 - 32 = \frac{9\mathrm{C}}{5}\)

\(45 = \frac{9\mathrm{C}}{5}\)

\(45 \times 5 = 9\mathrm{C}\)

\(225 = 9\mathrm{C}\)

\(\mathrm{C} = 25°\mathrm{C}\)

5. TRANSLATE the result back to inequality form

• The acceptable range is from \(20°\mathrm{C}\) to \(25°\mathrm{C}\), inclusive
• This translates to: \(20 \leq \mathrm{C} \leq 25\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misunderstand what the problem is asking and think they need to convert Celsius to Fahrenheit instead of Fahrenheit to Celsius, or they confuse which values represent the bounds.

This confusion leads them to either work with the wrong conversion direction or misinterpret the given temperature range. They may select Choice C (\(68 \leq \mathrm{C} \leq 77\)), thinking the Celsius range should match the Fahrenheit numbers.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors when solving for C, particularly when multiplying by 5 and dividing by 9, leading to incorrect temperature calculations.

Common calculation mistakes include getting C = 36 and C = 45 (by confusing intermediate steps), which leads them to select Choice B (\(36 \leq \mathrm{C} \leq 45\)).

The Bottom Line:

This problem requires careful attention to what's given versus what's asked for, combined with solid algebraic manipulation skills. The key insight is recognizing that you're converting FROM the given Fahrenheit range TO find the equivalent Celsius range.