A solid rectangular prism is made of a material with a uniform density of 4 grams per cubic centimeter. The...

1. TRANSLATE the problem information

• Given information:
  • Material density: 4 grams per cubic centimeter
  • Prism length: 10 centimeters
  • Prism width: 5 centimeters
  • Total mass: 600 grams
  • Need to find: height in centimeters

2. TRANSLATE the key relationship

• The fundamental relationship: \(\mathrm{Mass = Density × Volume}\)
• For a rectangular prism: \(\mathrm{Volume = length × width × height}\)
• Combined: \(\mathrm{Mass = Density × (length × width × height)}\)

3. SIMPLIFY by substitution and solving

• Substitute known values: \(\mathrm{600 = 4 × (10 × 5 × h)}\)
• Calculate the base area: \(\mathrm{600 = 4 × (50 × h)}\)
• Simplify: \(\mathrm{600 = 200h}\)
• Solve for height: \(\mathrm{h = 600 ÷ 200 = 3}\) centimeters

Answer: A (3)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students may get confused about which formula to use or how the given information connects to find the unknown.

Some students might try to use surface area formulas instead of volume, or might not recognize that they need to use the density-mass-volume relationship. This confusion about the fundamental approach leads to random guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the equation but make computational errors.

For example, they might calculate \(\mathrm{10 × 5 = 50}\) correctly, but then make errors like \(\mathrm{4 × 50 = 250}\) instead of 200, leading to \(\mathrm{h = 600 ÷ 250 = 2.4}\). Since 2.4 isn't an option, this causes confusion and potentially selecting the closest answer Choice A (3) by chance, or Choice B (12) through further miscalculation.

The Bottom Line:

This problem tests whether students can recognize density problems as fundamentally about the \(\mathrm{Mass = Density × Volume}\) relationship, then execute clean algebra to solve for the missing dimension.