The temperature y (in degrees Celsius) of a solution changes linearly with time x (in minutes). For every 3-minute increase...

1. TRANSLATE the problem information

• Given information:
  • Temperature changes linearly with time
  • For every 3-minute increase, temperature decreases by 5 degrees
  • When \(\mathrm{x = 8}\) minutes, temperature = 17°C
  • Need temperature at \(\mathrm{x = 0}\) minutes

• What this tells us: We need a linear function \(\mathrm{y = mx + b}\)

2. TRANSLATE the rate of change into slope

• 'For every 3-minute increase, temperature decreases by 5 degrees'
• This means: \(\mathrm{Δx = +3, Δy = -5}\) (negative because it decreases)
• Therefore: slope \(\mathrm{m = \frac{Δy}{Δx} = \frac{-5}{3}}\)

3. INFER how to find the y-intercept

• We have: \(\mathrm{y = \frac{-5}{3}x + b}\)
• We need to find \(\mathrm{b}\) using the known point \(\mathrm{(8, 17)}\)
• The temperature at \(\mathrm{x = 0}\) will be the y-intercept \(\mathrm{b}\)

4. SIMPLIFY to solve for b

• Substitute the point \(\mathrm{(8, 17)}\):
  \(\mathrm{17 = \frac{-5}{3}(8) + b}\)
• Calculate:
  \(\mathrm{17 = \frac{-40}{3} + b}\)
• Solve for \(\mathrm{b}\):
  \(\mathrm{b = 17 + \frac{40}{3}}\)
• Convert 17 to thirds:
  \(\mathrm{b = \frac{51}{3} + \frac{40}{3} = \frac{91}{3}}\) (use calculator)

5. Find the temperature at x = 0

• At \(\mathrm{x = 0}\):
  \(\mathrm{y = \frac{-5}{3}(0) + \frac{91}{3} = \frac{91}{3}}\)

Answer: \(\mathrm{\frac{91}{3}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Incorrectly interpreting the rate of change description. Students might think 'decreases by 5 for every 3-minute increase' means the slope is positive \(\mathrm{+\frac{5}{3}}\) (focusing on the increase in time) or might flip the fraction to get \(\mathrm{\frac{-3}{5}}\) (thinking change in time over change in temperature).

With slope = \(\mathrm{+\frac{5}{3}}\): Using point \(\mathrm{(8, 17)}\) gives \(\mathrm{17 = \frac{5}{3}(8) + b}\), so \(\mathrm{b = 17 - \frac{40}{3} = \frac{51}{3} - \frac{40}{3} = \frac{11}{3}}\).

This may lead them to select Choice A \(\mathrm{(\frac{11}{3})}\).

Second Most Common Error:

Poor SIMPLIFY execution: Getting the slope correct as \(\mathrm{\frac{-5}{3}}\) but making arithmetic errors with fractions when finding \(\mathrm{b}\). Students might incorrectly calculate \(\mathrm{17 + \frac{40}{3}}\) or make errors converting between mixed numbers and improper fractions.

Common mistakes include getting \(\mathrm{b = 27}\) or \(\mathrm{b = 32}\), which correspond to the other answer choices.

The Bottom Line:

This problem tests whether students can accurately translate verbal descriptions of rates into mathematical slope values. The fraction arithmetic adds another layer of complexity where calculation errors can derail an otherwise correct approach.