*(Note: Minimal change applied - question was already excellent)*A chocolatier pours melted chocolate into molds shaped like right triangular prisms....

1. TRANSLATE the problem information

• Given information:
  • Right triangular prism with length = \(15\) cm
  • Triangular base has longer leg = \(\mathrm{x}\) cm
  • Shorter leg = \(\mathrm{x - 2}\) cm (2 less than longer leg)
  • Need to find volume function \(\mathrm{V(x)}\)

2. INFER the approach

• For any prism: Volume = (area of base) × (height of prism)
• Since our base is a right triangle, we need the right triangle area formula
• The "height of prism" here is the given length of 15 cm

3. INFER and apply the right triangle area formula

• Area of right triangle = \(\frac{1}{2} \times \mathrm{leg_1} \times \mathrm{leg_2}\)
• Area = \(\frac{1}{2} \times \mathrm{x} \times (\mathrm{x - 2})\)

4. SIMPLIFY to find the volume function

• Volume = (area of base) × (length of prism)
• \(\mathrm{V(x)} = \left[\frac{1}{2} \times \mathrm{x} \times (\mathrm{x - 2})\right] \times 15\)
• \(\mathrm{V(x)} = \frac{15\mathrm{x}(\mathrm{x - 2})}{2}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students misinterpret "2 centimeters less than the longer leg" and write the shorter leg as \((\mathrm{x + 2})\) instead of \((\mathrm{x - 2})\).

This leads them to calculate the area as \(\frac{1}{2} \times \mathrm{x} \times (\mathrm{x + 2})\), giving volume \(\mathrm{V(x)} = \frac{15\mathrm{x}(\mathrm{x + 2})}{2}\).

This may lead them to select Choice B (\(\frac{15\mathrm{x}(\mathrm{x + 2})}{2}\)).

Second Most Common Error:

Poor INFER execution: Students forget that the volume formula for a prism requires multiplying by the length (15 cm) and only calculate the area of the triangular base.

This gives them \(\mathrm{V(x)} = \frac{1}{2} \times \mathrm{x} \times (\mathrm{x - 2}) = \frac{\mathrm{x}(\mathrm{x - 2})}{2}\), omitting the factor of 15 entirely.

This may lead them to select Choice D (\(\frac{\mathrm{x}(\mathrm{x - 2})}{2}\)).

The Bottom Line:

This problem tests whether students can systematically translate word relationships into algebra and then properly combine multiple geometric formulas. The key insight is recognizing that "prism volume" requires both the base area calculation AND multiplication by the prism's length.