A linear model estimates the monthly operating cost for a company from January 2020 to January 2025. The model estimates...

1. TRANSLATE the problem information

• Given information:
  • January 2020: 85 thousand dollars
  • January 2023: 145 thousand dollars
  • January 2025: z thousand dollars (unknown)
  • This is a linear model (straight line relationship)

2. INFER the coordinate approach

• Since we have a linear relationship, we can set up coordinates where:
  • x-axis = months after January 2020
  • y-axis = cost in thousands of dollars
• We need to find the slope first, then use it to find z

3. TRANSLATE dates to time coordinates

• January 2020: \(\mathrm{t = 0}\) months
• January 2023: \(\mathrm{t = 36}\) months (3 years × 12 months/year)
• January 2025: \(\mathrm{t = 60}\) months (5 years × 12 months/year)

4. SIMPLIFY the slope calculation

• Using points \(\mathrm{(0, 85)}\) and \(\mathrm{(36, 145)}\):
• Slope = \(\mathrm{\frac{145 - 85}{36 - 0}}\)
  \(\mathrm{= \frac{60}{36}}\)
  \(\mathrm{= \frac{5}{3}}\) thousand dollars per month

5. INFER the linear equation

• Using point-slope form with (0, 85):
• \(\mathrm{Cost = \frac{5}{3}t + 85}\)

6. SIMPLIFY to find z

• For January 2025, substitute \(\mathrm{t = 60}\):
• \(\mathrm{Cost = \frac{5}{3}(60) + 85}\)
  \(\mathrm{= 100 + 85}\)
  \(\mathrm{= 185}\) thousand dollars

Answer: C) 185

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret the time periods or incorrectly convert years to months. They might think January 2023 is 3 months after January 2020 instead of 36 months, or similarly miscalculate the time to January 2025.

This leads to an incorrect slope calculation and ultimately the wrong answer. This may cause them to select any of the incorrect answer choices or lead to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the problem but make arithmetic errors when calculating the slope \(\mathrm{\frac{60}{36} = \frac{5}{3}}\) or when multiplying \(\mathrm{\frac{5}{3} \times 60 = 100}\).

Common mistakes include getting \(\mathrm{\frac{3}{5}}\) instead of \(\mathrm{\frac{5}{3}}\) for the slope, or incorrectly calculating the final multiplication. This may lead them to select Choice A (165) or Choice B (175).

The Bottom Line:

This problem requires careful attention to time unit conversions and systematic application of the slope formula. Students who rush through the setup or make computational errors will struggle to reach the correct answer.