A right square prism has a height of 14 units. The volume of the prism is 2,016 cubic units. What...

1. TRANSLATE the problem information

• Given information:
  • Right square prism with height \(\mathrm{h = 14}\) units
  • Volume \(\mathrm{V = 2,016}\) cubic units
  • Need to find: side length s of the square base

• This tells us we need to use the volume formula for a right square prism

2. TRANSLATE the volume relationship

• Volume formula for a right square prism: \(\mathrm{V = s^2h}\)
  • \(\mathrm{s^2}\) represents the area of the square base
  • \(\mathrm{h}\) represents the height of the prism

3. SIMPLIFY by substituting known values

• Substitute \(\mathrm{V = 2,016}\) and \(\mathrm{h = 14}\) into the formula:

\(\mathrm{2,016 = s^2 \times 14}\)

4. SIMPLIFY to solve for s²

• Divide both sides by 14:

\(\mathrm{s^2 = 2,016 \div 14 = 144}\) (use calculator)

5. SIMPLIFY to solve for s

• Take the square root of both sides:

\(\mathrm{s = \pm\sqrt{144} = \pm12}\)

6. APPLY CONSTRAINTS to select the valid solution

• Since s represents a length measurement, it must be positive
• Therefore: \(\mathrm{s = 12}\) units

Answer: 12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may confuse this with other prism types or forget that a 'square prism' means the base is a square, not that the entire prism is a cube.

They might try to use \(\mathrm{V = lwh}\) thinking they need three different measurements, leading to confusion when they realize they only have two values (volume and height). This causes them to get stuck and abandon systematic solution, potentially guessing randomly.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students correctly solve the algebra to get \(\mathrm{s = \pm12}\), but then either select the negative value or get confused about which solution to choose.

Without recognizing that length measurements must be positive in real-world contexts, they might second-guess their answer or think the problem has no solution. This leads to confusion and guessing.

The Bottom Line:

This problem tests whether students can connect the specific language of 'right square prism' to the correct volume formula and then follow through with proper algebraic technique while applying real-world constraints to their mathematical solutions.