Let \(\mathrm{r(x) = (x + 5)(40 - x)}\). Define \(\mathrm{s(x) = r(x) - 12}\). For all real numbers x, what...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{r(x) = (x + 5)(40 - x)}\)
  • \(\mathrm{s(x) = r(x) - 12}\)
  • Need to find maximum value of \(\mathrm{s(x)}\)
• What this tells us: We need to first find the maximum of \(\mathrm{r(x)}\), then subtract 12

2. INFER the approach

• Since \(\mathrm{s(x)}\) is just \(\mathrm{r(x)}\) shifted down by 12 units, the maximum of \(\mathrm{s(x)}\) occurs at the same x-value as \(\mathrm{r(x)}\)
• To find maximum of \(\mathrm{r(x)}\), we need to recognize it's a quadratic function and find its vertex
• Strategy: Expand \(\mathrm{r(x)}\) into standard form, then use vertex formula

3. SIMPLIFY to find the quadratic form

• Expand \(\mathrm{r(x) = (x + 5)(40 - x)}\):
  \(\mathrm{r(x) = 40x - x^2 + 200 - 5x = -x^2 + 35x + 200}\)
• This is a quadratic with \(\mathrm{a = -1, b = 35, c = 200}\)
• Since \(\mathrm{a \lt 0}\), the parabola opens downward and has a maximum

4. APPLY vertex formula to find maximum

• Vertex x-coordinate: \(\mathrm{x = -b/(2a) = -35/(2(-1)) = 35/2}\)
• Maximum value of \(\mathrm{r(x)}\):
  \(\mathrm{r(35/2) = -(35/2)^2 + 35(35/2) + 200}\)
  \(\mathrm{= -1225/4 + 1225/2 + 200}\)
  \(\mathrm{= 2025/4}\)

5. APPLY CONSTRAINTS to find s(x) maximum

• Since \(\mathrm{s(x) = r(x) - 12}\):
  \(\mathrm{Maximum\ of\ s(x) = 2025/4 - 12 = 2025/4 - 48/4 = 1977/4}\)

Answer: D) 1977/4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak APPLY CONSTRAINTS reasoning: Students correctly find that \(\mathrm{r(x)}\) has maximum value 2025/4, but forget to subtract 12 to get the maximum of \(\mathrm{s(x)}\).

They see \(\mathrm{r_{max} = 2025/4}\) and think "that's my answer!" without remembering that the question asks for the maximum of \(\mathrm{s(x) = r(x) - 12}\), not \(\mathrm{r(x)}\) itself.

This leads them to select Choice E (2025/4).

Second Most Common Error:

Poor INFER strategy: Students might try to substitute the vertex \(\mathrm{x = 35/2}\) into the original factored form incorrectly, getting confused with the arithmetic and making computational errors.

Without a clear strategic approach, they may make calculation mistakes that lead to other incorrect answer choices or abandon systematic solution and guess.

The Bottom Line:

This problem tests whether students can systematically work through a multi-step optimization problem while keeping track of function transformations. The trap answer E (2025/4) specifically targets students who solve most of the problem correctly but miss the final transformation step.