A rectangular painting has an area of 72 square feet. If the width of the painting is 6 feet, what...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{Area} = 72\text{ square feet}\)
  • \(\mathrm{Width} = 6\text{ feet}\)
  • Need to find: Perimeter

• What this tells us: We have area and one dimension, but need both dimensions to find perimeter

2. INFER the solution approach

• Since perimeter requires both length and width, but we only know width, we must find length first
• Use the area formula to find the missing length dimension
• Then use both dimensions in the perimeter formula

3. SIMPLIFY to find the length

• \(\mathrm{Area} = \mathrm{length} \times \mathrm{width}\)
• \(72 = \mathrm{length} \times 6\)
• \(\mathrm{length} = 72 \div 6 = 12\text{ feet}\)

4. SIMPLIFY to find the perimeter

• \(\mathrm{Perimeter} = 2 \times (\mathrm{length} + \mathrm{width})\)
• \(\mathrm{Perimeter} = 2 \times (12 + 6)\)
• \(\mathrm{Perimeter} = 2 \times 18 = 36\text{ feet}\)

Answer: C) 36

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly find the length as 12 feet, but then forget the "multiply by 2" part of the perimeter formula.

They calculate: \(\mathrm{Perimeter} = \mathrm{length} + \mathrm{width} = 12 + 6 = 18\)

This may lead them to select Choice A (18).

Second Most Common Error:

Poor INFER reasoning: Students recognize they need to find perimeter but don't realize they need to find the length first.

They might try to use the given information directly, getting confused about how area relates to perimeter, leading to guessing or attempting incorrect calculations.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This is a two-step problem requiring students to work backwards from area to find a missing dimension, then forwards to perimeter. The most critical step is remembering that perimeter involves adding ALL sides - for a rectangle, that means 2 times the sum of length and width.