An isosceles trapezoid has a height of 8 centimeters. The length of the shorter base of the trapezoid is b...

1. TRANSLATE the problem information

• Given information:
  • Height: 8 cm
  • Shorter base: b cm
  • Longer base: "6 centimeters less than twice the length of the shorter base"
  • Need: Area function in terms of b

• The key translation: "6 less than twice the shorter base" = \(\mathrm{2b - 6}\)

2. INFER what formula to use

• We have a trapezoid area problem, so we'll use: \(\mathrm{A = \frac{1}{2}(b_1 + b_2)h}\)
• We need to substitute our known values: \(\mathrm{h = 8, b_1 = b, b_2 = 2b - 6}\)

3. SIMPLIFY by substituting and combining

• Start with: \(\mathrm{A = \frac{1}{2}(b_1 + b_2)h}\)
• Substitute: \(\mathrm{A = \frac{1}{2}(b + (2b - 6)) \times 8}\)
• Combine like terms inside parentheses: \(\mathrm{b + 2b - 6 = 3b - 6}\)
• So: \(\mathrm{A = \frac{1}{2}(3b - 6) \times 8}\)

4. SIMPLIFY the final expression

• Multiply the constants: \(\mathrm{\frac{1}{2} \times 8 = 4}\)
• Final function: \(\mathrm{A(b) = 4(3b - 6)}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misinterpreting "6 centimeters less than twice the length of the shorter base"

Students often write this as \(\mathrm{6 - 2b}\) instead of \(\mathrm{2b - 6}\). The phrase "6 less than [something]" means you subtract 6 FROM that something, not subtract that something from 6. With the wrong translation, they get:

• Longer base = \(\mathrm{6 - 2b}\)
• \(\mathrm{A = \frac{1}{2}(b + (6 - 2b)) \times 8}\)
  \(\mathrm{= \frac{1}{2}(6 - b) \times 8}\)
  \(\mathrm{= 4(6 - b)}\)

This doesn't match any answer choice exactly, leading to confusion and guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Making arithmetic errors during algebraic manipulation

Students might correctly translate to get \(\mathrm{2b - 6}\), but then make errors like:

• Forgetting to distribute properly: writing \(\mathrm{A = \frac{1}{2}(3b - 6) \times 8}\)
  \(\mathrm{= 4(3b) - 4(6)}\)
  \(\mathrm{= 12b - 24}\)
• Or incorrectly combining: \(\mathrm{b + 2b - 6 = 2b - 6}\) (dropping the b term)

These errors can lead them to select Choice D (\(\mathrm{8(3b - 6)}\)) if they use 8 instead of 4 as the coefficient.

The Bottom Line:

This problem tests whether students can accurately translate English phrases into mathematical expressions. The word order in "6 less than twice..." is especially tricky because students must reverse the order when writing the math expression.