The mass of an object is the product of its density and its volume. A certain object consists of two...

1. TRANSLATE the problem information

• Given information:
  • Object has two adjacent cubes
  • Larger cube side length = \(3 \times\) smaller cube side length
  • Uniform density = \(29.00 \text{ g/cm}^3\)
  • Total mass = \(4,872\) grams
• We need: Mass of the larger cube

2. INFER the solution approach

• Since we know total mass and density, we can find total volume
• Then we can work backwards to find individual cube volumes
• Strategy: Use a variable for the smaller cube's side length

3. TRANSLATE relationships into algebra

Let \(\mathrm{s}\) = side length of smaller cube (in cm)

• Volume of smaller cube = \(\mathrm{s}^3\) cm³
• Volume of larger cube = \((3\mathrm{s})^3 = 27\mathrm{s}^3\) cm³
• Total volume = \(\mathrm{s}^3 + 27\mathrm{s}^3 = 28\mathrm{s}^3\) cm³

4. INFER the key equation

• Since mass = density × volume:

\(29.00 \times (\text{total volume}) = 4,872\)

• Substituting: \(29.00 \times 28\mathrm{s}^3 = 4,872\)

5. SIMPLIFY to solve for s³

\(812\mathrm{s}^3 = 4,872\)

\(\mathrm{s}^3 = 4,872 \div 812 = 6\)

6. INFER the final calculation path

\(\text{Volume of larger cube} = 27\mathrm{s}^3 = 27 \times 6 = 162\) cm³

\(\text{Mass of larger cube} = 162 \times 29.00 = 4,698\) grams (use calculator)

Answer: 4,698 grams

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to convert "3 times the side length" into the volume relationship. They might think the larger cube has volume \(3\mathrm{s}^3\) instead of \((3\mathrm{s})^3 = 27\mathrm{s}^3\).

Using volume \(3\mathrm{s}^3\), they get total volume = \(\mathrm{s}^3 + 3\mathrm{s}^3 = 4\mathrm{s}^3\), leading to:

\(29.00 \times 4\mathrm{s}^3 = 4,872\), so \(\mathrm{s}^3 = 42\)

This gives larger cube volume = \(3 \times 42 = 126\) cm³

Mass = \(126 \times 29.00 = 3,654\) grams

This leads to confusion since 3,654 doesn't match the expected answer.

Second Most Common Error:

Incomplete SIMPLIFY execution: Students correctly set up \(29.00 \times 28\mathrm{s}^3 = 4,872\) but make arithmetic errors in the division or final multiplication.

Some might incorrectly calculate \(\mathrm{s}^3 = 6.5\) instead of 6, or make errors in \(162 \times 29.00\), leading to answers that are close but wrong.

The Bottom Line:

The critical insight is recognizing that when linear dimensions scale by factor 3, volumes scale by factor \(3^3 = 27\). Students who miss this cubic relationship will struggle from the very beginning.