A wildlife biologist models a lake's fish population with the function \(\mathrm{P(t) = c - 12t^2}\), where \(\mathrm{P(t)}\) represents the...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{P(t) = c - 12t^2}\)
  • Two months after spill: population = 3000 fish
  • Need to find: when population reaches zero

• Mathematical translation: \(\mathrm{P(2) = 3000}\)

2. INFER the solution approach

• We have one unknown constant c and one condition \(\mathrm{P(2) = 3000}\)
• Strategy: First find c, then solve for when \(\mathrm{P(t) = 0}\)
• This requires substitution followed by setting up a new equation

3. SIMPLIFY to find the constant c

Substitute \(\mathrm{t = 2}\) into the function:

\(\mathrm{P(2) = c - 12(2)^2}\)
\(\mathrm{3000 = c - 12(4)}\)
\(\mathrm{3000 = c - 48}\)
\(\mathrm{c = 3048}\)

4. TRANSLATE the zero population condition

• "Population reaches zero" means \(\mathrm{P(t) = 0}\)
• Our complete function is now \(\mathrm{P(t) = 3048 - 12t^2}\)

5. SIMPLIFY to solve for t

Set \(\mathrm{P(t) = 0}\):

\(\mathrm{0 = 3048 - 12t^2}\)
\(\mathrm{12t^2 = 3048}\)
\(\mathrm{t^2 = 254}\)
\(\mathrm{t = \sqrt{254} \approx 15.9}\) (use calculator)

Answer: C (15.9)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misinterpret "Two months after the spill, population is 3000" and try to use \(\mathrm{t = 3000}\) instead of \(\mathrm{P(2) = 3000}\), or confuse which value represents time versus population.

This fundamental misunderstanding prevents them from correctly finding c, leading to an entirely wrong function and causing them to guess among the answer choices.

Second Most Common Error:

Poor INFER reasoning: Students recognize they need \(\mathrm{P(2) = 3000}\) but don't realize they must find c first before solving \(\mathrm{P(t) = 0}\). They may try to work directly with the equation \(\mathrm{0 = c - 12t^2}\), not understanding they need the complete numerical function.

This incomplete approach leads to confusion and may cause them to select Choice A (12.0) if they incorrectly assume some relationship with the coefficient 12.

The Bottom Line:

This problem tests whether students can handle a two-stage process: using given information to complete a function, then using that complete function to answer the main question. Success requires careful translation of conditions into mathematical statements.