Question:A rectangular billboard has a perimeter of 192 inches. The length of the billboard is twice its width. What is...

1. TRANSLATE the problem information

• Given information:
  • Perimeter = 192 inches
  • Length = twice the width
  • Need to find: Area
• What this tells us:
  • If width = w, then length = 2w
  • We can use these in the perimeter formula

2. TRANSLATE the perimeter relationship into an equation

• Perimeter formula: \(\mathrm{P = 2(length + width)}\)
• Substituting our values: \(\mathrm{192 = 2(2w + w)}\)

3. SIMPLIFY the equation to find the width

• \(\mathrm{192 = 2(2w + w)}\)
• \(\mathrm{192 = 2(3w)}\)
• \(\mathrm{192 = 6w}\)
• \(\mathrm{w = 32}\) inches

4. INFER what to calculate next

• Now that we have the width, we need the length to find area
• Length = \(\mathrm{2w = 2(32) = 64}\) inches

5. SIMPLIFY the area calculation

• \(\mathrm{Area = length \times width}\)
• \(\mathrm{Area = 64 \times 32 = 2048}\) square inches (use calculator)

Answer: 2048

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students may set up the relationship incorrectly, writing width = 2 × length instead of length = 2 × width, or they may forget to properly substitute both expressions into the perimeter formula.

This leads to incorrect equations like \(\mathrm{192 = 2(w + 2w)}\) being solved as \(\mathrm{192 = 6w}\), which gives \(\mathrm{w = 32}\), but then they incorrectly think this is the length, leading to \(\mathrm{area = 32 \times 16 = 512}\).

Second Most Common Error:

Poor INFER execution: Students correctly find \(\mathrm{w = 32}\) but then calculate the area as \(\mathrm{32 \times 32 = 1024}\) because they forget that this width value needs to be used to find the length first.

They stop the solution process too early and don't recognize that finding one dimension requires calculating the other before finding area.

The Bottom Line:

This problem requires careful attention to the relationship between dimensions and systematic completion of all steps. Students who rush through without clearly defining their variables or who stop solving after finding just one dimension will struggle to get the correct area.