The area A, in square meters, of a triangular banner is given by \(\mathrm{A = \frac{1}{2}b(b + 8)}\), where b...

1. TRANSLATE the problem information

• Given information:
  • Area formula: \(\mathrm{A = \frac{1}{2}b(b + 8)}\)
  • \(\mathrm{b}\) is the base length in meters
  • Need to find an expression for height in meters

2. INFER the key relationship

• Both the given expression and the standard triangle area formula represent the area of the same triangle
• Standard triangle area formula: \(\mathrm{A = \frac{1}{2} \times base \times height = \frac{1}{2}bh}\)
• Since both equal A, we can set them equal: \(\mathrm{\frac{1}{2}b(b + 8) = \frac{1}{2}bh}\)

3. SIMPLIFY to solve for height

• Divide both sides by \(\mathrm{\frac{1}{2}b}\) (valid since \(\mathrm{b \neq 0}\) for a real triangle):
  • Left side: \(\mathrm{\frac{1}{2}b(b + 8) \div \frac{1}{2}b = (b + 8)}\)
  • Right side: \(\mathrm{\frac{1}{2}bh \div \frac{1}{2}b = h}\)
• Therefore: \(\mathrm{h = b + 8}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they should set the given expression equal to the standard area formula. Instead, they might try to manipulate the given expression \(\mathrm{A = \frac{1}{2}b(b + 8)}\) directly without connecting it to \(\mathrm{A = \frac{1}{2}bh}\). This leads to confusion about how to extract height information from the expression, causing them to get stuck and guess.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misidentify what represents the height in the given expression. They might think that since the expression is \(\mathrm{\frac{1}{2}b(b + 8)}\), the height must be just one of the factors like \(\mathrm{b}\), leading them to select Choice A (b).

The Bottom Line:

This problem tests whether students can connect a specific area expression to the general triangle area formula and use that connection to extract the height component.