A right square pyramid has a volume of 2/3 cubic meters and a height of 8 meters. What is the...

1. TRANSLATE the problem information

• Given information:
  • Right square pyramid with volume = \(\frac{2}{3}\) cubic meters
  • Height = \(8\) meters
  • Need to find: side length of square base

• What this tells us: We have V and h, need to find s using the pyramid volume relationship

2. INFER the approach

• Since we know volume and height but need the base dimension, we should use the volume formula for a pyramid
• For a square pyramid: \(\mathrm{V} = \frac{1}{3}\mathrm{s}^2\mathrm{h}\), where s is the side length of the square base
• We can substitute our known values and solve for s

3. SIMPLIFY by setting up and solving the equation

• Substitute into \(\mathrm{V} = \frac{1}{3}\mathrm{s}^2\mathrm{h}\):

\(\frac{2}{3} = \frac{1}{3}\mathrm{s}^2(8)\)

• Simplify the right side:

\(\frac{2}{3} = \frac{8}{3}\mathrm{s}^2\)

• Multiply both sides by 3/8 to isolate s²:

\(\mathrm{s}^2 = \frac{2}{3} \times \frac{3}{8} = \frac{6}{24} = \frac{1}{4}\)

• Take the square root of both sides:

\(\mathrm{s} = \sqrt{\frac{1}{4}} = \frac{1}{2}\)

Answer: B (1/2)

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge: Volume formula for pyramid
Students may not remember that pyramid volume uses \(\mathrm{V} = \frac{1}{3}\mathrm{Bh}\) rather than \(\mathrm{V} = \mathrm{Bh}\) (which applies to prisms). They might set up: \(\frac{2}{3} = \mathrm{s}^2(8)\), leading to \(\mathrm{s}^2 = \frac{2}{3} \div 8 = \frac{2}{24} = \frac{1}{12}\), and \(\mathrm{s} = \sqrt{\frac{1}{12}} \approx 0.29\), which doesn't match any answer choice. This leads to confusion and guessing.

Second Most Common Error:

Weak SIMPLIFY execution: Fraction arithmetic errors
Students correctly set up \(\frac{2}{3} = \frac{8}{3}\mathrm{s}^2\) but make mistakes when solving for s². They might incorrectly compute \(\frac{2}{3} \times \frac{3}{8}\) as \(\frac{6}{11}\) instead of \(\frac{6}{24} = \frac{1}{4}\), or make errors taking the square root. Common wrong calculations lead them to select Choice A (1/4) by confusing \(\mathrm{s}^2 = \frac{1}{4}\) with \(\mathrm{s} = \frac{1}{4}\).

The Bottom Line:

This problem requires precise recall of the pyramid volume formula and careful fraction arithmetic. Students who rush through the algebra or confuse pyramid formulas with prism formulas will struggle to reach the correct answer systematically.