The density of a certain type of wood is 353 kilograms per cubic meter. A sample of this type of...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{Density = 353\,kg/m^3}\)
  • \(\mathrm{Mass = 345\,kg}\)
  • Sample is cube-shaped
  • Need edge length to nearest hundredth of a meter

2. INFER the solution approach

• Strategy: Use the density relationship to find volume first, then use cube geometry to find the edge length
• Since we know density and mass, we can find volume using: \(\mathrm{Density = \frac{Mass}{Volume}}\)

3. SIMPLIFY to find the volume

• Set up the density equation: \(\mathrm{353 = \frac{345}{V}}\)
• Solve for volume: \(\mathrm{V = \frac{345}{353}\,m^3}\)
• Calculate: \(\mathrm{V ≈ 0.9773\,m^3}\) (use calculator)

4. INFER the cube relationship

• For a cube with edge length s: \(\mathrm{Volume = s^3}\)
• So we have: \(\mathrm{s^3 = \frac{345}{353}}\)

5. SIMPLIFY to find the edge length

• Take the cube root: \(\mathrm{s = \sqrt[3]{\frac{345}{353}}}\)
• Calculate: \(\mathrm{s = \sqrt[3]{0.9773} ≈ 0.9924\,m}\) (use calculator)
• Round to nearest hundredth: \(\mathrm{s = 0.99\,m}\)

Answer: B. 0.99

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may confuse the density formula setup, writing \(\mathrm{345 = \frac{353}{V}}\) instead of \(\mathrm{353 = \frac{345}{V}}\). This gives them \(\mathrm{V = \frac{353}{345} ≈ 1.023}\), leading to \(\mathrm{s = \sqrt[3]{1.023} ≈ 1.008}\), which rounds to 1.01.

This may lead them to select Choice C (1.01).

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up all relationships but make calculation errors when finding the cube root or rounding. They might get \(\mathrm{s ≈ 0.984}\) and incorrectly round to 0.98, or get \(\mathrm{s ≈ 1.016}\) and round to 1.02.

This causes them to select Choice A (0.98) or Choice D (1.02).

The Bottom Line:

This problem requires careful attention to the density formula setup and precise calculation of cube roots. The key insight is recognizing that you need volume first, then can use cube geometry to find the edge length.