A right square pyramid has a volume of 24 cubic centimeters and a height of 6 centimeters. What is the...

1. TRANSLATE the problem information

• Given information:
  • Volume = 24 cubic centimeters
  • Height = 6 centimeters
  • Need to find: side length of square base

2. INFER the approach needed

• We need the volume formula for a pyramid: \(\mathrm{V = \frac{1}{3}Bh}\)
• For a square base, the base area \(\mathrm{B = s^2}\) where s is the side length
• We can substitute our known values and solve for s

3. TRANSLATE into mathematical equation

• Set up: \(\mathrm{24 = \frac{1}{3}(s^2)(6)}\)

4. SIMPLIFY the equation step by step

• First, multiply out the right side: \(\mathrm{24 = \frac{1}{3} \times s^2 \times 6 = 2s^2}\)
• Divide both sides by 2: \(\mathrm{s^2 = 12}\)
• Take the square root: \(\mathrm{s = \sqrt{12}}\)

5. SIMPLIFY the radical

• Factor out perfect squares: \(\mathrm{\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}}\)

Answer: D (\(\mathrm{2\sqrt{3}}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students correctly set up the equation but make calculation errors or don't know how to simplify \(\mathrm{\sqrt{12}}\).

They might leave their answer as \(\mathrm{\sqrt{12}}\) and select Choice B (\(\mathrm{\sqrt{12}}\)), not realizing this needs to be simplified to match the answer format. Or they might incorrectly simplify \(\mathrm{\sqrt{12}}\), perhaps thinking \(\mathrm{\sqrt{12} = \sqrt{4} + \sqrt{3} = 2 + \sqrt{3}}\), leading to confusion and guessing.

Second Most Common Error:

Missing conceptual knowledge: Students don't remember that for a square base, the area is \(\mathrm{s^2}\), not just s.

They might set up \(\mathrm{24 = \frac{1}{3}(s)(6)}\), getting \(\mathrm{s = 12}\) and then guessing among the radical options, possibly selecting Choice D (\(\mathrm{2\sqrt{3}}\)) by luck or selecting Choice C (\(\mathrm{2\sqrt{2}}\)) through similar reasoning.

The Bottom Line:

This problem requires both proper formula setup and careful radical simplification. Students need to recognize that \(\mathrm{\sqrt{12}}\) must be simplified to match the answer choices, and that perfect square factors can be extracted from under the radical.