An auditorium has seats for 1,800 people. Tickets to attend a show at the auditorium currently cost $4.00. For each...

1. TRANSLATE the problem information

• Given equation: \(\mathrm{y = -100x^2 + 1,400x + 7,200}\)
• This represents revenue (y) based on ticket price increase (x)
• We need to find the x-value where maximum revenue occurs

2. INFER the mathematical approach

• This is a quadratic function in standard form \(\mathrm{y = ax^2 + bx + c}\)
• Since the coefficient of \(\mathrm{x^2}\) is -100 (negative), the parabola opens downward
• For a downward-opening parabola, the vertex represents the maximum point
• We need to find the x-coordinate of the vertex

3. TRANSLATE the coefficients from the equation

• From \(\mathrm{y = -100x^2 + 1,400x + 7,200}\):
  • \(\mathrm{a = -100}\)
  • \(\mathrm{b = 1,400}\)
  • \(\mathrm{c = 7,200}\)

4. INFER the correct formula to use

• For any quadratic \(\mathrm{y = ax^2 + bx + c}\), the x-coordinate of the vertex is: \(\mathrm{x = \frac{-b}{2a}}\)
• This formula will give us the x-value where the maximum occurs

5. SIMPLIFY using the vertex formula

• \(\mathrm{x = \frac{-b}{2a}}\)
• \(\mathrm{x = \frac{-1,400}{2(-100)}}\)
• \(\mathrm{x = \frac{-1,400}{-200}}\)
• \(\mathrm{x = \frac{1,400}{200}}\)
• \(\mathrm{x = 7}\)

Answer: B. 7

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize this as a vertex-finding problem or confuse maximum vs. minimum. They might see the quadratic equation but not connect it to the need for finding where the maximum occurs. Some students might try to plug in the answer choices or attempt to solve \(\mathrm{y = 0}\), not realizing they need the vertex.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the need for the vertex formula but make arithmetic errors in the calculation. Common mistakes include:

• Getting confused with the double negative: \(\mathrm{\frac{-1,400}{2(-100)}}\) and calculating it as \(\mathrm{\frac{-1,400}{200} = -7}\)
• Sign errors when simplifying fractions with negatives
• Basic division errors like \(\mathrm{1,400 \div 200 = 70}\) instead of 7

This may lead them to select Choice D (18) if they make significant calculation errors, or get a negative result that doesn't match any choices.

The Bottom Line:

This problem tests whether students can connect the real-world context of "finding maximum revenue" to the mathematical concept of finding a parabola's vertex, then execute the vertex formula correctly without sign errors.