A rectangular volleyball court has an area of 162 square meters. If the length of the court is twice the...

1. TRANSLATE the problem information

• Given information:
  • Rectangular court with area = 162 square meters
  • Length = twice the width
• What we need to find: the width

2. TRANSLATE the relationships into mathematical expressions

• Let \(\mathrm{w}\) = width of the court
• Then \(\mathrm{length = 2w}\) (since length is twice the width)
• Area formula: \(\mathrm{A = length \times width}\)

3. INFER the solution approach

• We have area and a relationship between length and width
• We can substitute the length expression into the area formula
• This will give us one equation with one unknown (w)

4. SIMPLIFY by setting up and solving the equation

• Substitute into area formula: \(\mathrm{162 = (2w) \times w}\)
• This gives us: \(\mathrm{162 = 2w^2}\)
• Divide both sides by 2: \(\mathrm{w^2 = 81}\)
• Take the square root: \(\mathrm{w = ±9}\)
• Since width must be positive: \(\mathrm{w = 9\text{ meters}}\)

Answer: A. 9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students correctly solve for \(\mathrm{w = 9}\), but then think they need to find the length instead of the width, since 9 seems 'small' for a volleyball court.

They calculate \(\mathrm{length = 2w = 2(9) = 18}\) and select this as their answer.

This leads them to select Choice B (18).

Second Most Common Error:

Conceptual confusion about area vs. perimeter: Students mix up the area formula with perimeter, thinking that 'length plus width' should equal 162.

They set up: \(\mathrm{length + width = 162}\), so \(\mathrm{2w + w = 162}\), giving \(\mathrm{3w = 162}\), so \(\mathrm{w = 54}\).

This leads them to select Choice D (54).

The Bottom Line:

This problem requires careful attention to what quantity you're solving for and making sure you use the correct geometric formula. The key insight is recognizing that you're working with area (multiplication of dimensions) rather than perimeter (addition of dimensions).