A rectangular container has a square base. The height of the container is three times the side length of the...

1. TRANSLATE the problem information

• Given information:
  • Container has a square base (side length = \(\mathrm{s}\))
  • Height = \(3 \times \text{(side length)} = 3\mathrm{s}\)
  • Total surface area = 56 square meters
• Need to find: Volume in cubic meters

2. INFER the approach needed

• Strategy: Find the surface area formula for this specific container, then solve for \(\mathrm{s}\)
• Once we know \(\mathrm{s}\), we can calculate the volume using \(\mathrm{V} = \text{length} \times \text{width} \times \text{height}\)

3. INFER the surface area components

• Container has 6 faces:
  • Top square: area = \(\mathrm{s}^2\)
  • Bottom square: area = \(\mathrm{s}^2\)
  • Four rectangular sides: each has area = \(\mathrm{s} \times \text{height} = \mathrm{s} \times 3\mathrm{s} = 3\mathrm{s}^2\)

4. SIMPLIFY to find the total surface area formula

• Total surface area = \(2\mathrm{s}^2 + 4(3\mathrm{s}^2)\)
  \(= 2\mathrm{s}^2 + 12\mathrm{s}^2\)
  \(= 14\mathrm{s}^2\)

5. SIMPLIFY to solve for the side length

• Set up equation: \(14\mathrm{s}^2 = 56\)
• Divide both sides by 14: \(\mathrm{s}^2 = 4\)
• Take square root: \(\mathrm{s} = 2\) meters

6. SIMPLIFY to find the volume

• Height = \(3\mathrm{s} = 3(2) = 6\) meters
• Volume = \(\mathrm{s} \times \mathrm{s} \times \text{height}\)
  \(= 2 \times 2 \times 6\)
  \(= 24\) cubic meters

Answer: C. 24

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misunderstand what "total surface area" means and only calculate the area of one face or forget to include all six faces of the container.

For example, they might only calculate \(2\mathrm{s}^2\) (top and bottom) and ignore the four sides, leading to the equation \(2\mathrm{s}^2 = 56\), which gives \(\mathrm{s}^2 = 28\) and \(\mathrm{s} = \sqrt{28} \approx 5.3\). This produces an unrealistic volume that doesn't match any answer choice, causing confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(14\mathrm{s}^2 = 56\) but make calculation errors, such as getting \(\mathrm{s} = 14\) instead of \(\mathrm{s} = 2\), or they correctly find \(\mathrm{s} = 2\) but miscalculate the final volume (forgetting to multiply by the height properly).

This may lead them to select Choice A (8) if they calculate \(\mathrm{s}^2 \times \text{height}\) incorrectly as \(4 \times 2 = 8\) instead of \(4 \times 6 = 24\).

The Bottom Line:

This problem requires careful attention to all six faces of the three-dimensional container. Students who rush through the surface area calculation or make basic algebraic errors will struggle to reach the correct volume.