The population of a certain species of bird on an island, t years after a conservation program begins, is modeled...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{P(t) = (200 + 10t)(4 - 0.2t)}\)
  • Time: 10 years after program begins means \(\mathrm{t = 10}\)
  • Need: \(\mathrm{P(10)}\)

2. SIMPLIFY by substituting the value

• Substitute \(\mathrm{t = 10}\) into both parts of the function:

\(\mathrm{P(10) = (200 + 10(10))(4 - 0.2(10))}\)

3. SIMPLIFY each parenthetical expression separately

• First parenthesis: \(\mathrm{200 + 10(10) = 200 + 100 = 300}\)
• Second parenthesis: \(\mathrm{4 - 0.2(10) = 4 - 2 = 2}\)

4. SIMPLIFY by multiplying the results

• \(\mathrm{P(10) = 300 \times 2 = 600}\)

Answer: C. 600

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly substitute \(\mathrm{t = 10}\) but make arithmetic errors or stop partway through the calculation.

Some students calculate only the first factor \(\mathrm{(200 + 100 = 300)}\) and stop there, thinking that's the answer. This incomplete solution leads them to select Choice A (300).

Second Most Common Error Path:

Poor TRANSLATE reasoning: Students partially substitute the value, applying \(\mathrm{t = 10}\) to only one part of the expression rather than both.

For example, they might calculate \(\mathrm{(200 + 10)(4 - 0.2(10)) = 210 \times 2 = 420}\), forgetting to substitute \(\mathrm{t = 10}\) into the first factor. This leads them to select Choice B (420).

The Bottom Line:

Function evaluation problems require careful, systematic substitution throughout the entire expression. Students must resist the urge to shortcut the process and ensure they've completed all algebraic steps before selecting their answer.