A right rectangular prism has a height of 9 inches. The length of the prism's base is x inches, which...

1. TRANSLATE the problem information

• Given information:
  • Height of prism = 9 inches
  • Length of base = x inches
  • "Length is 7 inches more than width"

• What this tells us: If length is 7 more than width, then width must be 7 less than length

2. TRANSLATE the dimensional relationship

• Since length = x and length is 7 more than width:
  • Width = x - 7 inches
• Now we have all dimensions in terms of x:
  • Length = x inches
  • Width = (x - 7) inches
  • Height = 9 inches

3. INFER the volume formula approach

• For any rectangular prism: Volume = length × width × height
• We need to substitute our expressions for each dimension

4. SIMPLIFY to find the volume function

• \(\mathrm{V(x) = length \times width \times height}\)
• \(\mathrm{V(x) = x \times (x - 7) \times 9}\)
• \(\mathrm{V(x) = 9x(x - 7)}\)

Answer: D. \(\mathrm{V(x) = 9x(x - 7)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misinterpreting "length is 7 inches more than width" as meaning width = x + 7

Students often confuse which variable should have the +7 and which should have the -7. If they think the width is x + 7, they get:

\(\mathrm{V(x) = x \times (x + 7) \times 9 = 9x(x + 7)}\)

This may lead them to select Choice C (\(\mathrm{9x(x + 7)}\))

Second Most Common Error:

Poor INFER reasoning: Treating height as a variable instead of the given constant 9

Some students see multiple variables in the answer choices and incorrectly assume height should also be expressed as (x + 9) or (x - 7), leading to expressions like x(x + 9)(x + 7) or x(x + 9)(x - 7).

This may lead them to select Choice A (\(\mathrm{x(x + 9)(x + 7)}\)) or Choice B (\(\mathrm{x(x + 9)(x - 7)}\))

The Bottom Line:

The key challenge is carefully translating the English phrase about the relationship between dimensions into the correct algebraic expressions. Students must be precise about which dimension is larger and express the smaller dimension accordingly.