A right rectangular pyramid has a height of 10 centimeters. The base is a rectangle whose length is x centimeters,...

1. TRANSLATE the problem information

• Given information:
  • Right rectangular pyramid with \(\mathrm{height = 10\text{ cm}}\)
  • Base rectangle has \(\mathrm{length = x\text{ cm}}\)
  • Base rectangle has width = "4 centimeters less than its length"

• TRANSLATE that phrase: \(\mathrm{width = x - 4\text{ cm}}\)

2. INFER what formula to use

• We need to find volume of a pyramid
• Pyramid volume formula: \(\mathrm{V = \frac{1}{3} \times (base\text{ }area) \times (height)}\)
• Since the base is rectangular: \(\mathrm{base\text{ }area = length \times width}\)

3. SIMPLIFY by setting up the calculation

• Base area \(\mathrm{= x \times (x - 4) = x(x - 4)}\)
• Height \(\mathrm{= 10}\)
• Volume \(\mathrm{= \frac{1}{3} \times x(x - 4) \times 10}\)

4. SIMPLIFY the final expression

• \(\mathrm{V(x) = \frac{1}{3} \times 10 \times x(x - 4)}\)
• \(\mathrm{V(x) = \frac{10}{3}x(x - 4)}\)

Answer: (B) \(\mathrm{V(x) = \frac{10}{3}x(x - 4)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misinterpreting "4 centimeters less than its length" as meaning \(\mathrm{x + 4}\) instead of \(\mathrm{x - 4}\).

Students sometimes think "less than" means addition rather than subtraction, or they get confused about the direction of the relationship. They might think "4 less than x" means "4 + x" instead of "x - 4".

This leads them to calculate base area as \(\mathrm{x(x + 4)}\) instead of \(\mathrm{x(x - 4)}\), and they select Choice (D) \(\mathrm{V(x) = \frac{10}{3}x(x + 4)}\).

Second Most Common Error:

Missing conceptual knowledge: Forgetting that pyramid volume includes the factor \(\mathrm{\frac{1}{3}}\), not just base area × height.

Students remember that volume involves base area times height but forget the \(\mathrm{\frac{1}{3}}\) factor that distinguishes pyramid volume from prism volume. They calculate \(\mathrm{V(x) = 10 \times x(x - 4) = 10x(x - 4)}\).

This leads them to select Choice (C) \(\mathrm{V(x) = 10x(x - 4)}\).

The Bottom Line:

This problem tests your ability to translate verbal relationships into algebraic expressions while correctly applying the pyramid volume formula. The key challenge is carefully interpreting "less than" relationships and remembering the \(\mathrm{\frac{1}{3}}\) factor that makes pyramids different from prisms.