A cup of coffee is cooling in a room that is at a constant temperature of 70 degrees Fahrenheit. The...

1. TRANSLATE the problem information

• Given information:
  • Room temperature: constant \(70°\mathrm{F}\)
  • Temperature difference model: \(30 \times (\frac{2}{3})^\mathrm{t}\)
  • t = time in minutes after coffee was poured
  • Need to find: y-intercept of coffee temperature graph

• What this tells us: The model gives us the difference between coffee temperature and room temperature, not the actual coffee temperature.

2. INFER what the y-intercept represents

• The y-intercept occurs when \(\mathrm{t} = 0\) (initial time when coffee was poured)
• To find the actual coffee temperature, we need: \(\mathrm{Coffee\ temp} = \mathrm{Room\ temp} + \mathrm{Temperature\ difference}\)
• Strategy: Evaluate the difference at \(\mathrm{t} = 0\), then add room temperature

3. SIMPLIFY the exponential expression at t = 0

• Temperature difference at \(\mathrm{t} = 0\): \(30 \times (\frac{2}{3})^0\)
• Since any non-zero number raised to the 0 power equals 1: \((\frac{2}{3})^0 = 1\)
• Therefore: \(30 \times 1 = 30°\mathrm{F}\) difference

4. INFER the initial coffee temperature

• Coffee temperature = Room temperature + Temperature difference
• Coffee temperature = \(70 + 30 = 100°\mathrm{F}\)
• Y-intercept = \((0, 100)\)

Answer: C. (0, 100)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students incorrectly evaluate \((\frac{2}{3})^0\)

Many students think \((\frac{2}{3})^0 = 0\) because they confuse the exponent rule with multiplication by zero. This gives them a temperature difference of \(30 \times 0 = 0°\mathrm{F}\), leading to a coffee temperature of \(70 + 0 = 70°\mathrm{F}\).

This may lead them to select Choice A: (0, 70).

Second Most Common Error:

Poor TRANSLATE reasoning: Students mistake what the model represents

Some students think the expression \(30 \times (\frac{2}{3})^\mathrm{t}\) directly gives the coffee temperature, not realizing it represents the temperature difference. At \(\mathrm{t} = 0\), they get \(30°\mathrm{F}\) and think this is the coffee temperature.

This causes them to get stuck since 30 isn't an answer choice, leading to confusion and guessing.

The Bottom Line:

This problem requires careful attention to what the mathematical model actually represents (temperature difference, not absolute temperature) and solid knowledge of exponential function properties.