A manufacturing company produces electronic components at a facility with a daily capacity of 12,000 units. The total production cost...

1. INFER the problem strategy

• Given information:
  • Cost function: \(\mathrm{C = 5x^2 - 60x + 320}\)
  • Need to find minimum cost
  • x represents thousands of components (constraint: \(\mathrm{0 \leq x \leq 12}\))

• This is a quadratic function, so finding the minimum means finding the vertex of the parabola

2. INFER the method for finding minimum

• Since the coefficient of \(\mathrm{x^2}\) is positive (\(\mathrm{a = 5}\)), the parabola opens upward
• The vertex represents the minimum point
• Use the vertex formula: \(\mathrm{x = -b/(2a)}\)

3. SIMPLIFY using the vertex formula

• Identify coefficients: \(\mathrm{a = 5, b = -60, c = 320}\)
• Apply formula: \(\mathrm{x = -b/(2a)}\)
  \(\mathrm{x = -(-60)/(2 \times 5)}\)
  \(\mathrm{x = 60/10}\)
  \(\mathrm{x = 6}\)

4. APPLY CONSTRAINTS to verify the answer

• Check that \(\mathrm{x = 6}\) falls within the given range \(\mathrm{0 \leq x \leq 12}\) ✓
• The solution is valid

Answer: B (6)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize this as a vertex-finding problem and instead try to substitute answer choices or use calculus methods they haven't mastered yet.

Without the strategic insight that quadratic optimization problems require finding the vertex, students may randomly test values or get overwhelmed by the algebraic expression. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the need for the vertex formula but make arithmetic errors, particularly with the double negative in \(\mathrm{-(-60)}\).

Common mistakes include calculating \(\mathrm{-(-60)/(2 \times 5)}\) as \(\mathrm{-60/10 = -6}\) (forgetting the negative sign cancellation) or making division errors like \(\mathrm{60/10 = 5}\). This may lead them to select Choice A (5) or get frustrated and guess.

The Bottom Line:

This problem requires recognizing the connection between "minimization" and "vertex of parabola" - a conceptual leap that many students miss, leading them to overcomplicate the solution or avoid systematic approaches entirely.