\(\mathrm{f(\theta) = -0.28(\theta - 27)^2 + 880}\). An engineer wanted to identify the best angle for a cooling fan in...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(\theta) = -0.28(\theta - 27)^2 + 880}\) models airflow vs. angle
  • Need to find the angle θ that gives greatest airflow

• What this tells us: We need to find the maximum value of this quadratic function

2. INFER the mathematical approach

• This function is already in vertex form: \(\mathrm{f(x) = a(x - h)^2 + k}\)
• The vertex form immediately shows us the maximum/minimum point
• Since we want "greatest airflow," we need to find where the function reaches its maximum

3. INFER the vertex location and type

• From \(\mathrm{f(\theta) = -0.28(\theta - 27)^2 + 880}\), identify the vertex form components:
  • \(\mathrm{a = -0.28}\) (coefficient of squared term)
  • \(\mathrm{h = 27}\) (value inside parentheses)
  • \(\mathrm{k = 880}\) (constant term)

• The vertex is at \(\mathrm{(h, k) = (27, 880)}\)

4. INFER whether this vertex is a maximum or minimum

• Since \(\mathrm{a = -0.28}\) is negative, the parabola opens downward
• A downward-opening parabola has its maximum at the vertex
• Therefore, the maximum airflow occurs at \(\mathrm{\theta = 27}\) degrees

Answer: C. 27

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the vertex form structure or confuse what the vertex represents.

Many students see the numbers -0.28, 27, and 880 but don't understand which value answers the question. They might think the coefficient (-0.28) or the maximum value (880) is the angle that produces maximum airflow, rather than recognizing that 27 is the input value (angle) that produces the maximum output (airflow).

This may lead them to select Choice A (-0.28) or Choice D (880).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand what "greatest airflow" means mathematically.

Some students might not connect "greatest airflow" with finding the maximum of the function. Without this connection, they may randomly select values from the equation without understanding the optimization aspect of the problem.

This leads to confusion and guessing among the available choices.

The Bottom Line:

This problem requires recognizing vertex form and understanding that optimization problems (finding greatest/least values) connect directly to the vertex of quadratic functions. Students who haven't internalized the vertex form pattern will struggle to identify which number represents the optimal input value.