A triangular pennant has a height of h inches. The base of the triangle is 3 inches more than twice...

1. TRANSLATE the problem information

• Given information:
  • Height of triangle: \(\mathrm{h}\) inches
  • Base description: "3 inches more than twice the height"

• TRANSLATE the base description into algebra:
  • "twice the height" = \(\mathrm{2h}\)
  • "3 inches more than twice the height" = \(\mathrm{2h + 3}\) inches

2. INFER the approach needed

• We need to find \(\mathrm{A(h)}\), a function giving area in terms of height
• Since we have expressions for both base and height, we can use the triangle area formula
• Strategy: Substitute our expressions into \(\mathrm{A = \frac{1}{2} \times base \times height}\)

3. APPLY the triangle area formula

• Triangle area formula: \(\mathrm{A = \frac{1}{2} \times base \times height}\)
• Substitute our values:
  • Base = \(\mathrm{(2h + 3)}\)
  • Height = \(\mathrm{h}\)
• \(\mathrm{A(h) = \frac{1}{2} \times (2h + 3) \times h}\)

4. SIMPLIFY to match answer choice format

• \(\mathrm{A(h) = \frac{1}{2}(2h + 3)(h)}\)
• Rearrange: \(\mathrm{A(h) = \frac{h(2h + 3)}{2}}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misinterpreting "3 inches more than twice the height"

Students often confuse the order and translate this as \(\mathrm{(2h - 3)}\) instead of \(\mathrm{(2h + 3)}\). The phrase "3 more than twice the height" means you start with twice the height \(\mathrm{(2h)}\) and then add 3, giving \(\mathrm{(2h + 3)}\). When students get this backwards, they work with the wrong base expression throughout.

This may lead them to select Choice A (\(\mathrm{A(h) = \frac{h(2h - 3)}{2}}\))

Second Most Common Error:

Missing conceptual knowledge: Forgetting the factor of 1/2 in triangle area formula

Some students remember that triangle area involves base times height but forget the 1/2 factor, leading to \(\mathrm{A(h) = h(2h + 3)}\) instead of \(\mathrm{A(h) = \frac{h(2h + 3)}{2}}\).

This may lead them to select Choice C (\(\mathrm{A(h) = h(2h + 3)}\))

The Bottom Line:

This problem tests your ability to translate wordy descriptions into precise algebraic expressions. The key insight is recognizing that "3 more than twice the height" follows the pattern "3 + 2h" which we write as \(\mathrm{(2h + 3)}\), not the other way around.