The table above shows the population of Greenleaf, Idaho, for the years 2000 and 2010. If the relationship between population...

1. TRANSLATE the problem information

• Given information:
  • Year 2000: Population = 862
  • Year 2010: Population = 846
  • Linear relationship
  • \(\mathrm{t}\) = years after 2000 (so \(\mathrm{t = 0}\) means year 2000, \(\mathrm{t = 10}\) means year 2010)

2. INFER the function structure

• Since the relationship is linear, we need \(\mathrm{P(t) = mt + b}\)
• We need to find slope \(\mathrm{(m)}\) and y-intercept \(\mathrm{(b)}\)
• The y-intercept occurs when \(\mathrm{t = 0}\), which corresponds to year 2000

3. Find the y-intercept (b)

• When \(\mathrm{t = 0}\) (year 2000), \(\mathrm{P(0) = 862}\)
• Therefore, \(\mathrm{b = 862}\)

4. SIMPLIFY to find the slope

• We have two points: \(\mathrm{(0, 862)}\) and \(\mathrm{(10, 846)}\)
• Slope = \(\mathrm{\frac{846 - 862}{10 - 0} = \frac{-16}{10} = -1.6}\)

5. Write the final function

• \(\mathrm{P(t) = -1.6t + 862 = 862 - 1.6t}\)

Answer: A. \(\mathrm{P(t) = 862 - 1.6t}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students calculate the slope as \(\mathrm{\frac{846 - 862}{10 - 0} = \frac{-16}{10}}\) but forget to reduce the fraction, leaving the slope as -16 instead of -1.6.

This leads them to write \(\mathrm{P(t) = 862 - 16t}\) and select Choice B (\(\mathrm{P(t) = 862 - 16t}\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret what "t years after 2000" means and think t represents the actual year (like \(\mathrm{t = 2000, t = 2010}\)) rather than the number of years elapsed since 2000.

This confusion leads them to work with expressions involving \(\mathrm{(t - 2000)}\) and select Choice C (\(\mathrm{P(t) = 862 + 16(t - 2,000)}\)) or Choice D (\(\mathrm{P(t) = 862 - 1.6(t - 2,000)}\)).

The Bottom Line:

This problem tests whether students can correctly set up a linear model by understanding the time variable definition and accurately computing slope through fraction reduction.