A small business models its daily profit, \(\mathrm{P(x)}\), in dollars, from selling x handmade scarves. The function \(\mathrm{P(x) = 0.25x(x...

1. TRANSLATE the problem information

• Given information:
  • Profit function: \(\mathrm{P(x) = 0.25x(x - 90)}\)
  • Need to find when profit = $0
  • Answer must be non-zero

• This means we need to solve: \(\mathrm{P(x) = 0}\)

2. TRANSLATE to set up the equation

• Set the profit function equal to zero:
  \(\mathrm{0.25x(x - 90) = 0}\)

3. INFER the solution strategy

• This is a product equal to zero
• We can use the zero product property: if \(\mathrm{a \times b = 0}\), then either \(\mathrm{a = 0}\) or \(\mathrm{b = 0}\)
• The constant \(\mathrm{0.25 \neq 0}\), so either \(\mathrm{x = 0}\) or \(\mathrm{(x - 90) = 0}\)

4. SIMPLIFY to find all solutions

• Case 1: \(\mathrm{x = 0}\)
• Case 2: \(\mathrm{x - 90 = 0}\)
  Adding 90 to both sides: \(\mathrm{x = 90}\)

• Both \(\mathrm{x = 0}\) and \(\mathrm{x = 90}\) make the profit equal to $0

5. APPLY CONSTRAINTS to select final answer

• The problem specifically asks for the NON-ZERO number of scarves
• Therefore, we select \(\mathrm{x = 90}\)

Answer: D) 90

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak APPLY CONSTRAINTS reasoning: Students find both solutions (\(\mathrm{x = 0}\) and \(\mathrm{x = 90}\)) but fail to notice that the question asks specifically for the "non-zero" solution.

They see that \(\mathrm{x = 0}\) gives zero profit and think "that makes sense - if you sell zero scarves, you make zero profit" without reading the constraint carefully. This leads them to select Choice B (0).

Second Most Common Error:

Poor INFER skill with zero product property: Students see \(\mathrm{0.25x(x - 90) = 0}\) but don't recognize they can use the zero product property.

Instead, they might try to expand: \(\mathrm{0.25x^2 - 22.5x = 0}\), then get confused with the quadratic formula or factoring the more complex expression. This leads to calculation errors and guessing among the remaining choices.

The Bottom Line:

This problem tests whether students can systematically find zeros of a factored function and then carefully read what the question is actually asking for. The mathematical work is straightforward, but the constraint about "non-zero" is the key detail that separates correct from incorrect answers.