The area of a triangle is 240 square inches. If the height of the triangle is 30 inches, what is...

1. TRANSLATE the problem information

• Given information:
  • Area of triangle = 240 square inches
  • Height = 30 inches
  • Need to find: length of the base
• This tells us we need to use the triangle area formula and solve for the missing base.

2. TRANSLATE the approach into mathematical setup

• Use the triangle area formula: \(\mathrm{A = \frac{1}{2}bh}\)
• Substitute the known values: \(\mathrm{240 = \frac{1}{2} \times b \times 30}\)

3. SIMPLIFY the equation step by step

• First, simplify the right side: \(\mathrm{\frac{1}{2} \times 30 = 15}\)
• The equation becomes: \(\mathrm{240 = 15b}\)
• Divide both sides by 15: \(\mathrm{b = 240 \div 15}\)
• Calculate: \(\mathrm{b = 16}\)

Answer: 16 inches (Choice C)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may confuse which variable represents what, or forget that the area formula requires \(\mathrm{\frac{1}{2}}\) as the coefficient.

Some students might write \(\mathrm{A = bh}\) instead of \(\mathrm{A = \frac{1}{2}bh}\), leading them to calculate:

\(\mathrm{240 = b \times 30}\), so \(\mathrm{b = 8}\)

This may lead them to select Choice B (8).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the equation but make arithmetic errors in the division step.

They might incorrectly calculate \(\mathrm{240 \div 15}\), perhaps getting confused with the steps and arriving at \(\mathrm{240 \div 30 = 8}\) or making other calculation mistakes.

This causes them to get stuck or select an incorrect answer choice.

The Bottom Line:

This problem tests whether students can correctly recall and apply the triangle area formula, then execute clean algebraic manipulation. The most critical step is remembering that triangle area includes the \(\mathrm{\frac{1}{2}}\) factor.