Between 2:00 PM and 8:00 PM, the temperature T, in degrees Fahrenheit, increases at a constant rate. At 2:00 PM,...

1. TRANSLATE the problem information

• Given information:
  • At 2:00 PM: \(\mathrm{T = 70°F}\)
  • At 8:00 PM: \(\mathrm{T = 88°F}\)
  • Temperature increases at constant rate
  • Need temperature at 5:00 PM

2. INFER the mathematical approach

• Since we have a constant rate, this is a linear relationship
• We need to set up coordinates and find the slope
• Let \(\mathrm{t}\) = hours after 2:00 PM as our time variable

3. TRANSLATE times into coordinates

• 2:00 PM → \(\mathrm{t = 0, T = 70}\)
• 8:00 PM → \(\mathrm{t = 6}\) (6 hours later), \(\mathrm{T = 88}\)
• 5:00 PM → \(\mathrm{t = 3}\) (3 hours after 2:00 PM), \(\mathrm{T = ?}\)

4. SIMPLIFY by calculating the rate (slope)

• Slope = (change in temperature)/(change in time)
• Slope = \(\frac{88 - 70}{6 - 0}\) = \(\frac{18}{6}\) = \(\mathrm{3°F}\) per hour

5. INFER the linear equation

• Using point \(\mathrm{(0, 70)}\) and slope = \(\mathrm{3}\):
• \(\mathrm{T = 70 + 3t}\)

6. SIMPLIFY to find temperature at 5:00 PM

• At \(\mathrm{t = 3}\):
  \(\mathrm{T = 70 + 3(3)}\)
  \(\mathrm{= 70 + 9}\)
  \(\mathrm{= 79°F}\)
• Alternative check: 5:00 PM is midway between 2:00 PM and 8:00 PM
• So \(\mathrm{T}\) = \(\frac{70 + 88}{2}\) = \(\mathrm{79°F}\) ✓

Answer: C) 79

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret the time intervals, particularly confusing "8:00 PM" as "8 hours" instead of recognizing it's 6 hours after 2:00 PM.

This leads to using an incorrect time span (8 instead of 6) when calculating slope: \(\frac{88-70}{8}\) = \(\frac{18}{8}\) = \(\mathrm{2.25°F}\) per hour. Then at "5 hours": \(\mathrm{T = 70 + 2.25(5) = 81.25°F}\), which doesn't match any answer choice exactly, causing confusion and potentially leading them to select Choice D (80) as the closest option.

Second Most Common Error:

Poor INFER reasoning: Students recognize the time intervals correctly but fail to realize they need to establish a systematic coordinate approach. Instead, they try to work directly with clock times, getting confused about what mathematical operations to perform.

This leads to attempts like trying to find "how much per hour" by dividing 18°F by some unclear time measure, or making up ratios that don't correspond to the actual problem structure. This causes them to get stuck and guess randomly among the choices.

The Bottom Line:

This problem requires converting a real-world scenario into mathematical coordinates. The key insight is recognizing that clock times must be converted to elapsed time intervals, and then applying linear function concepts systematically.