\(\mathrm{P(n) = -5(n - k)^2 + 405}\)A company's daily profit P(n), in thousands of dollars, is modeled by the function...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{P(n) = -5(n - k)^2 + 405}\) (profit function)
  • When \(\mathrm{n = 5}\) employees, profit = \(\mathrm{\$280{,}000 = 280}\) thousand dollars
  • \(\mathrm{k \gt 5}\) (constraint on k)
  • Need to find n where profit = \(\mathrm{\$0}\) (break-even)
  • This break-even point satisfies \(\mathrm{n \gt k}\)

2. SIMPLIFY to find the value of k

• Substitute the known condition \(\mathrm{P(5) = 280}\):
  \(\mathrm{280 = -5(5 - k)^2 + 405}\)

• Solve step by step:
  \(\mathrm{-125 = -5(5 - k)^2}\)
  \(\mathrm{25 = (5 - k)^2}\)
  \(\mathrm{\pm 5 = 5 - k}\)

• This gives us two possibilities: \(\mathrm{k = 0}\) or \(\mathrm{k = 10}\)

3. APPLY CONSTRAINTS to determine k

• Since \(\mathrm{k \gt 5}\), we must have \(\mathrm{k = 10}\)

4. TRANSLATE the break-even condition

• Break-even means profit = 0, so set \(\mathrm{P(n) = 0}\):
  \(\mathrm{0 = -5(n - 10)^2 + 405}\)

5. SIMPLIFY to solve for n

• Rearrange the equation:
  \(\mathrm{5(n - 10)^2 = 405}\)
  \(\mathrm{(n - 10)^2 = 81}\)

• Take the square root of both sides:
  \(\mathrm{n - 10 = \pm 9}\)

• This gives us \(\mathrm{n = 19}\) or \(\mathrm{n = 1}\)

6. APPLY CONSTRAINTS to select the final answer

• We need \(\mathrm{n \gt k = 10}\)
• Since \(\mathrm{19 \gt 10}\) but \(\mathrm{1 \lt 10}\), the answer is \(\mathrm{n = 19}\)

Answer: D (19)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skills: Students may misinterpret "\(\mathrm{\$280{,}000}\)" as 280,000 instead of 280 (thousands of dollars), leading to incorrect equation setup: \(\mathrm{280{,}000 = -5(5 - k)^2 + 405}\). This creates an impossible equation since the maximum value of the function is 405 thousand dollars. This leads to confusion and guessing.

Second Most Common Error:

Inadequate APPLY CONSTRAINTS execution: Students correctly find \(\mathrm{k = 10}\) and the two break-even points \(\mathrm{n = 1}\) and \(\mathrm{n = 19}\), but forget to apply the constraint \(\mathrm{n \gt k}\). They might select the first solution they calculate. This may lead them to select Choice A (1).

The Bottom Line:

This problem tests whether students can work systematically through a multi-step quadratic application while carefully tracking units and applying multiple constraints. The key insight is recognizing that both finding k and finding the break-even point involve solving quadratic equations, and each step requires applying given constraints to select the correct solution.