A small business sells handmade bracelets. The owner determines that the weekly profit, \(\mathrm{P(x)}\), in dollars, can be modeled by...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{P(x) = -1.5(x - 30)^2 + 1200}\) represents weekly profit
  • \(\mathrm{x}\) = price of one bracelet in dollars
  • Need to find: price that gives greatest weekly profit

• What this tells us: We need to find the value of \(\mathrm{x}\) that maximizes \(\mathrm{P(x)}\)

2. INFER the mathematical approach

• This is a quadratic function written in vertex form: \(\mathrm{y = a(x - h)^2 + k}\)
• For optimization problems with quadratics, the answer is always at the vertex
• We need to identify the vertex and determine what its coordinates mean

3. INFER the vertex location and meaning

• Comparing \(\mathrm{P(x) = -1.5(x - 30)^2 + 1200}\) with \(\mathrm{y = a(x - h)^2 + k}\):
  • \(\mathrm{a = -1.5, h = 30, k = 1200}\)
• The vertex is at point \(\mathrm{(h, k) = (30, 1200)}\)
• Since \(\mathrm{a = -1.5 \lt 0}\), the parabola opens downward, making this vertex a maximum point

4. TRANSLATE the vertex coordinates back to context

• Vertex coordinates \(\mathrm{(30, 1200)}\) mean:
  • When price = $30 (x-coordinate), the profit = $1200 (y-coordinate)
  • This is the maximum possible profit
• The question asks for the optimal price, which is the x-coordinate: 30

Answer: B) 30

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that the vertex coordinates have different meanings - confusing which coordinate answers the question.

Students correctly identify the vertex as \(\mathrm{(30, 1200)}\) but then select the y-coordinate (1200) because it's the larger number or because they think "greatest profit" means they want the profit value itself, not the price that creates it.

This may lead them to select Choice D) 1200.

Second Most Common Error:

Missing conceptual knowledge about vertex form: Students don't recognize the function as being in vertex form or don't remember that \(\mathrm{(h, k)}\) represents the vertex.

Without this recognition, they might try to find the maximum by setting the derivative to zero or by plugging in the answer choices, both of which are unnecessarily complex approaches that can lead to computational errors.

This leads to confusion and guessing among the available choices.

The Bottom Line:

This problem tests whether students can connect the mathematical concept of a vertex to its real-world meaning in an optimization context. The key insight is that for "find the input that maximizes the output" problems, you want the x-coordinate of the vertex, not the y-coordinate.