Question:The function \(\mathrm{C(r) = 0.8\pi r^2}\) gives the cost, in dollars, to apply a specific type of liquid fertilizer to...

1. TRANSLATE the function notation

• Given: \(\mathrm{C(40) = 1,280π}\) and the function \(\mathrm{C(r) = 0.8πr^2}\)
• In function notation \(\mathrm{f(x) = y}\):
  • The value inside parentheses (x) is the input
  • The value after the equals sign (y) is the output

2. TRANSLATE the specific values

• For \(\mathrm{C(40) = 1,280π}\):
  • Input: 40 (this is the radius r in meters)
  • Output: \(\mathrm{1,280π}\) (this is the cost C(r) in dollars)

3. INFER the real-world meaning

• Since the function gives cost based on radius:
  • "When radius = 40 meters, cost = \(\mathrm{1,280π}\) dollars"
• This translates to: "If the radius of the circular plot is 40 meters, then the cost to apply the fertilizer is \(\mathrm{1,280π}\) dollars"

4. Match with answer choices

• (A) Matches our interpretation exactly ✓
• (B), (C), (D) all contain errors in interpreting the function notation

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse which number represents the input versus the output in function notation.

They might think "40 looks smaller, so maybe it's the output" or get confused about which variable goes where. This leads them to interpret \(\mathrm{C(40) = 1,280π}\) as "when the cost is 40 dollars, the radius is \(\mathrm{1,280π}\) meters" or similar reversed logic.

This may lead them to select Choice C (reversing input/output roles) or Choice D (swapping the numerical values).

Second Most Common Error:

Conceptual confusion about variables: Students correctly identify that 40 is the input but then misinterpret what the output represents.

Since they see \(\mathrm{1,280π}\) and know that area formulas often involve π, they might think "\(\mathrm{1,280π}\) must be the area" instead of recognizing it as the cost output from the given function.

This may lead them to select Choice B (\(\mathrm{1,280π}\) square meters as area).

The Bottom Line:

Function notation problems require careful attention to what goes in (input) versus what comes out (output). The key is remembering that \(\mathrm{f(input) = output}\), regardless of which numbers "look bigger" or more familiar.