A company's daily profit P, in thousands of dollars, is a function of the number of units, x, it produces....

1. TRANSLATE the problem information

• Given: \(\mathrm{P(x) = 200 - 0.5(x - 120)^2}\)
• Find: Number of units (x) that maximizes daily profit

2. INFER the mathematical structure

• This is a quadratic function in vertex form: \(\mathrm{f(x) = a(x - h)^2 + k}\)
• Since the coefficient is negative (\(\mathrm{-0.5}\)), the parabola opens downward
• A downward-opening parabola has a maximum value at its vertex

3. INFER the optimization strategy

• For vertex form \(\mathrm{f(x) = a(x - h)^2 + k}\), the extremum occurs at \(\mathrm{x = h}\)
• To maximize \(\mathrm{P(x) = 200 - 0.5(x - 120)^2}\), we need to minimize the term \(\mathrm{0.5(x - 120)^2}\)
• Since \(\mathrm{(x - 120)^2 \geq 0}\) always, its minimum value is 0

4. SIMPLIFY to find the maximum point

• Set the squared term equal to zero: \(\mathrm{(x - 120)^2 = 0}\)
• Taking the square root: \(\mathrm{x - 120 = 0}\)
• Solving: \(\mathrm{x = 120}\)

Answer: C) 120

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the vertex form structure and instead try to expand the quadratic or use calculus methods like taking derivatives.

They might expand: \(\mathrm{P(x) = 200 - 0.5(x^2 - 240x + 14400) = 200 - 0.5x^2 + 120x - 7200 = -0.5x^2 + 120x - 7000}\), then try to complete the square or use the quadratic formula unnecessarily. This leads to confusion and potential calculation errors, causing them to get stuck and guess.

Second Most Common Error:

Conceptual confusion about vertex form: Students might identify \(\mathrm{h = -120}\) instead of \(\mathrm{h = 120}\), misreading the sign in \(\mathrm{(x - 120)^2}\).

This may lead them to select Choice A (60) or get confused about which direction the shift goes.

The Bottom Line:

The key insight is recognizing that this problem tests understanding of vertex form, not complex algebraic manipulation. Students who immediately see the vertex form pattern can solve this in seconds, while those who don't recognize it get lost in unnecessary calculations.