A right pyramid with a rectangular base has a height of 9 inches. The length of the base is x...

1. TRANSLATE the problem information

• Given information:
  • Right pyramid with rectangular base
  • Height = 9 inches
  • Length of base = x inches
  • "Length is 7 inches more than width"

• What this tells us: If \(\mathrm{length = x}\), then \(\mathrm{width = x - 7}\)

2. INFER the solution approach

• To find volume, we need: \(\mathrm{V = \frac{1}{3} \times base\ area \times height}\)
• Since the base is rectangular, we need: \(\mathrm{base\ area = length \times width}\)
• Strategy: Find base area first, then calculate volume

3. Calculate the base area

• Base area = length × width = \(\mathrm{x(x - 7)}\)

4. SIMPLIFY to find the volume

• \(\mathrm{V(x) = \frac{1}{3} \times base\ area \times height}\)
• \(\mathrm{V(x) = \frac{1}{3} \times x(x - 7) \times 9}\)
• \(\mathrm{V(x) = 3x(x - 7)}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "length is 7 inches more than width" and incorrectly conclude that \(\mathrm{width = x + 7}\) instead of \(\mathrm{width = x - 7}\).

They think: "If length is 7 more than width, and length = x, then width must be x + 7." This backwards reasoning leads to \(\mathrm{base\ area = x(x + 7)}\), giving them \(\mathrm{V(x) = 3x(x + 7)}\).

This may lead them to select Choice C (\(\mathrm{V(x) = 3x(x + 7)}\))

The Bottom Line:

This problem tests careful translation of verbal relationships into mathematical expressions. The key challenge is correctly interpreting "A is 7 more than B" to mean \(\mathrm{A = B + 7}\), which implies \(\mathrm{B = A - 7}\).