A materials scientist studies two different alloy compositions and their relationship between temperature (in degrees Celsius) and internal pressure (...

1. TRANSLATE the problem requirements

• Given information:
  • Alloy A: \(\mathrm{P_A = 0.35T + 8.0}\)
  • Alloy B: \(\mathrm{P_B = 0.45T - 1.0}\)
  • Temperature: \(\mathrm{T = 150°C}\)
• What we need: "by how much does pressure in Alloy B exceed pressure in Alloy A"
• This means we need \(\mathrm{P_B - P_A}\) at \(\mathrm{T = 150}\)

2. SIMPLIFY to find \(\mathrm{P_A}\) at \(\mathrm{T = 150}\)

• \(\mathrm{P_A = 0.35T + 8.0}\)
• \(\mathrm{P_A = 0.35(150) + 8.0}\)
• \(\mathrm{P_A = 52.5 + 8.0}\)
• \(\mathrm{P_A = 60.5}\) atmospheres

3. SIMPLIFY to find \(\mathrm{P_B}\) at \(\mathrm{T = 150}\)

• \(\mathrm{P_B = 0.45T - 1.0}\)
• \(\mathrm{P_B = 0.45(150) - 1.0}\)
• \(\mathrm{P_B = 67.5 - 1.0}\)
• \(\mathrm{P_B = 66.5}\) atmospheres

4. SIMPLIFY to find the difference

• \(\mathrm{P_B - P_A = 66.5 - 60.5}\)
• \(\mathrm{P_B - P_A = 6.0}\) atmospheres

Answer: B. 6.0

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students might misinterpret "by how much does B exceed A" and calculate \(\mathrm{P_A - P_B}\) instead of \(\mathrm{P_B - P_A}\), getting -6.0. Since this isn't an answer choice, they may select the absolute value or guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Arithmetic errors in the decimal multiplications (\(\mathrm{0.35 × 150}\) or \(\mathrm{0.45 × 150}\)) or in the final subtraction. For example, miscalculating \(\mathrm{0.35 × 150}\) as 53.5 instead of 52.5, leading to \(\mathrm{P_A = 61.5}\) and a final difference of 5.0, causing them to select Choice A (5.0).

The Bottom Line:

This problem tests careful reading comprehension and systematic arithmetic execution. The mathematics is straightforward, but precision in both language interpretation and calculation is essential for success.