A function P assigns to a positive number x the perimeter, in units, of a square whose area is x...

1. TRANSLATE the problem information

• Given information:
  • Function P assigns perimeter to a positive number x
  • The square has area x square units
  • We need to find \(\mathrm{P(169)}\)

2. INFER the mathematical relationships

• To find perimeter, we need the side length of the square
• Since \(\mathrm{area = x}\) and \(\mathrm{area = side^2}\), the side length must be \(\sqrt{\mathrm{x}}\)
• The perimeter of any square is 4 times its side length

3. TRANSLATE this into function form

• \(\mathrm{P(x) = 4 \times (side\ length) = 4\sqrt{x}}\)

4. SIMPLIFY for the specific value

• \(\mathrm{P(169) = 4\sqrt{169}}\)
• \(\sqrt{169} = 13\)
• \(\mathrm{P(169) = 4 \times 13 = 52}\)

Answer: D (52)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misinterpret what the function P represents, thinking it directly relates to the number 169 rather than understanding that 169 represents the area of a square.

Some students might think \(\mathrm{P(169)}\) means "the perimeter of a square with side length 169" and calculate \(4 \times 169 = 676\). Since this isn't an answer choice, this leads to confusion and guessing.

Second Most Common Error:

Missing conceptual knowledge about area formula: Students who forget that \(\mathrm{area = side^2}\) may not realize they need to take the square root to find the side length.

They might try to work backwards incorrectly, perhaps dividing 169 by 4 to get 42.25, which doesn't match any choice. This may lead them to select Choice A (13) by incorrectly thinking the side length equals \(\sqrt{169} = 13\), which is actually the correct side length but not the perimeter.

The Bottom Line:

This problem tests whether students can work with function notation while connecting area and perimeter formulas for squares. The key insight is recognizing that the input to function P is the area, not the side length, requiring an extra step to find the side length first.