A residential pool installation company has safety regulations that restrict the dimensions of rectangular pools they can install in certain...

1. TRANSLATE the problem information

• Given information:
  • Pool is rectangular with \(\mathrm{length = 3 \times width}\)
  • Walkway around pool is 15 feet wide
  • Constraint: \(\mathrm{perimeter\, of\, pool + walkway\, width \leq 180\, feet}\)

• Let \(\mathrm{w = width\, of\, pool}\) (in feet)
• Then \(\mathrm{length = 3w}\) feet

2. INFER the mathematical setup

• We need to find the perimeter first, then apply the constraint
• Perimeter of rectangle = \(\mathrm{2(length + width)}\)
  \(\mathrm{= 2(3w + w)}\)
  \(\mathrm{= 2(4w)}\)
  \(\mathrm{= 8w}\) feet
• The constraint becomes: \(\mathrm{8w + 15 \leq 180}\)

3. SIMPLIFY by solving the inequality

\(\mathrm{8w + 15 \leq 180}\)
\(\mathrm{8w \leq 180 - 15}\)
\(\mathrm{8w \leq 165}\)
\(\mathrm{w \leq 165/8}\)

4. SIMPLIFY the fraction to mixed number form

• \(\mathrm{165 \div 8 = 20}\) remainder 5
• So \(\mathrm{165/8 = 20\frac{5}{8}}\)

5. APPLY CONSTRAINTS to complete the inequality

• Since width must be positive: \(\mathrm{0 \lt w \leq 20\frac{5}{8}}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make arithmetic errors when converting \(\mathrm{165/8}\) to a mixed number.

They might incorrectly calculate \(\mathrm{165 \div 8}\), getting results like \(\mathrm{20\frac{3}{8}}\) or \(\mathrm{20\frac{3}{4}}\) instead of \(\mathrm{20\frac{5}{8}}\). Some students struggle with long division or make computational errors.

This may lead them to select Choice A \(\mathrm{(16\frac{7}{8})}\) or Choice B (20) depending on their calculation mistake.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret what 'width of the walkway' means in the constraint.

They might think they need to calculate the perimeter of the walkway area instead of just using the 15-foot width value directly. This leads to setting up a much more complex equation or getting confused about the geometry.

This leads to confusion and abandoning systematic solution, causing them to guess.

The Bottom Line:

This problem tests whether students can cleanly translate a real-world constraint into algebra and then execute the arithmetic accurately. The key insight is recognizing that 'width of walkway' is simply the given value 15, not something that needs geometric calculation.