A thermostat in a greenhouse is set to maintain a temperature of 24.0 degrees Celsius (°C). The device is designed...

1. TRANSLATE the problem information

• Given information:
  • Set point temperature: \(24.0°\mathrm{C}\)
  • Tolerance: temperature stays 'within \(1.8°\mathrm{C}\) of the set point'

• What 'within \(1.8°\mathrm{C}\)' means: The actual temperature can be at most \(1.8°\mathrm{C}\) higher or \(1.8°\mathrm{C}\) lower than \(24.0°\mathrm{C}\)

2. SIMPLIFY to find the bounds

• Maximum allowable temperature:

\(24.0 + 1.8 = 25.8°\mathrm{C}\)

• Minimum allowable temperature:

\(24.0 - 1.8 = 22.2°\mathrm{C}\)

3. INFER how to represent the complete range

• The temperature t must satisfy BOTH conditions:
  • t must be greater than or equal to \(22.2°\mathrm{C}\)
  • t must be less than or equal to \(25.8°\mathrm{C}\)

• This creates a compound inequality: \(22.2°\mathrm{C} \leq \mathrm{t} \leq 25.8°\mathrm{C}\)

Answer: B) \(22.2°\mathrm{C} \leq \mathrm{t} \leq 25.8°\mathrm{C}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret 'within \(1.8°\mathrm{C}\) of \(24.0°\mathrm{C}\)' and think it means the range goes from \(1.8°\mathrm{C}\) to \(24.0°\mathrm{C}\), using the tolerance value as a boundary rather than understanding it as the amount of deviation allowed from the set point.

This leads them to select Choice A (\(1.8°\mathrm{C} \leq \mathrm{t} \leq 24.0°\mathrm{C}\)).

Second Most Common Error:

Poor INFER reasoning: Students correctly calculate the bounds (\(22.2°\mathrm{C}\) and \(25.8°\mathrm{C}\)) but fail to recognize that both bounds must be included simultaneously in a compound inequality. They might think the temperature can be either below \(22.2°\mathrm{C}\) OR above \(25.8°\mathrm{C}\).

This may lead them to select Choice C (\(\mathrm{t} \leq 22.2°\mathrm{C}\)) or Choice D (\(\mathrm{t} \geq 25.8°\mathrm{C}\)).

The Bottom Line:

The key challenge is translating the everyday phrase 'within X of Y' into mathematical language. Students must recognize that this creates a symmetric interval centered at Y, not a range starting from X.