A company's daily profit, \(\mathrm{P(n)}\), in dollars, from selling a certain product is given by the function \(\mathrm{P(n) = -4n^2...

1. TRANSLATE the problem information

• Given information:
  • Profit function: \(\mathrm{P(n) = -4n^2 + 160n - 750}\)
  • n = number of units sold \(\mathrm{(n \geq 0)}\)
  • Need to find maximum possible daily profit

2. INFER the mathematical approach

• This is a quadratic function in standard form \(\mathrm{(an^2 + bn + c)}\)
• Since the coefficient of \(\mathrm{n^2}\) is negative \(\mathrm{(-4)}\), the parabola opens downward
• A downward-opening parabola has a maximum value at its vertex
• Strategy: Find the vertex, then calculate the profit at that point

3. SIMPLIFY to find the optimal number of units

• Use the vertex formula: \(\mathrm{n = -b/(2a)}\)
• From \(\mathrm{P(n) = -4n^2 + 160n - 750}\), we have \(\mathrm{a = -4}\) and \(\mathrm{b = 160}\)
• Calculate: \(\mathrm{n = -160/(2 \times -4) = -160/(-8) = 20}\) units

4. SIMPLIFY to find the maximum profit

• Substitute \(\mathrm{n = 20}\) into the profit function:

\(\mathrm{P(20) = -4(20)^2 + 160(20) - 750}\)

• Calculate step by step:

\(\mathrm{P(20) = -4(400) + 3200 - 750}\)
\(\mathrm{P(20) = -1600 + 3200 - 750}\)
\(\mathrm{P(20) = 850}\)

Answer: C) 850

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not recognize this as an optimization problem requiring the vertex of a parabola. Instead, they might try to set \(\mathrm{P(n) = 0}\) to solve for break-even points, or attempt to use calculus methods they're not comfortable with. This leads to confusion and abandoning systematic solution, causing them to guess among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the need to find the vertex but make arithmetic errors. Common mistakes include:

• Calculating \(\mathrm{n = -160/(2 \times -4)}\) incorrectly (getting \(\mathrm{-20}\) instead of \(\mathrm{20}\))
• Making substitution errors when calculating \(\mathrm{P(20)}\), particularly with the sign of \(\mathrm{-1600}\)

These calculation errors typically lead them to select Choice A) 20 (confusing the optimal number of units with the maximum profit) or Choice B) 750 (from arithmetic mistakes in the final calculation).

The Bottom Line:

Success requires recognizing that profit optimization problems with quadratic functions always involve finding the vertex, combined with careful arithmetic execution through multiple steps.