A park ranger hung squirrel houses each in the shape of a right rectangular prism for fox squirrels. Each house...

1. TRANSLATE the problem information

• Given information:
  • Each house is a right rectangular prism
  • \(\mathrm{Height = 11\ inches}\)
  • \(\mathrm{Length\ of\ base = x\ inches}\)
  • Length is 1 inch more than the width

• What this tells us: We need to express width in terms of x

2. TRANSLATE the length-width relationship

• "Length is 1 inch more than width" means:
  • \(\mathrm{Length = Width + 1}\)
  • Since \(\mathrm{Length = x}\), we have: \(\mathrm{x = Width + 1}\)
  • Therefore: \(\mathrm{Width = x - 1}\)

3. INFER the volume calculation approach

• For any rectangular prism: \(\mathrm{Volume = length \times width \times height}\)
• We need to substitute our known values to get a function \(\mathrm{V(x)}\)

4. SIMPLIFY by substituting values

• \(\mathrm{V = length \times width \times height}\)
• \(\mathrm{V = x \times (x - 1) \times 11}\)
• \(\mathrm{V = 11x(x - 1)}\)

Answer: A. \(\mathrm{V(x) = 11x(x - 1)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misinterpreting "length is 1 inch more than width"

Students often think this means \(\mathrm{width = x + 1}\) instead of \(\mathrm{width = x - 1}\). They reverse the relationship, thinking "if length is more than width, then width must be x + 1." This fundamental translation error leads them to calculate \(\mathrm{V = 11x(x + 1)}\).

This leads them to select Choice B (\(\mathrm{V(x) = 11x(x + 1)}\))

The Bottom Line:

The key challenge is carefully translating the verbal relationship between length and width. The phrase "A is more than B" means \(\mathrm{A = B + (amount)}\), so when length is 1 more than width and \(\mathrm{length = x}\), we get \(\mathrm{x = width + 1}\), which means \(\mathrm{width = x - 1}\).