A right square pyramid has a height of 9 centimeters and a volume of 300 cubic centimeters. What is the...

1. TRANSLATE the problem information

• Given information:
  • Height = 9 centimeters
  • Volume = 300 cubic centimeters
  • Base is a square
• Need to find: Length of an edge of the base

2. INFER the approach

• Since we know volume and height, we can use the volume formula to find the base dimensions
• For a right square pyramid: \(\mathrm{V = \frac{1}{3} \times base\ area \times height}\)
• Since the base is square with edge length s: \(\mathrm{base\ area = s^2}\)

3. TRANSLATE this into an equation

• \(\mathrm{V = \frac{1}{3} \times s^2 \times h}\)
• Substitute known values: \(\mathrm{300 = \frac{1}{3} \times s^2 \times 9}\)

4. SIMPLIFY to solve for s

\(\mathrm{300 = \frac{1}{3} \times s^2 \times 9}\)

\(\mathrm{300 = 3s^2}\)

\(\mathrm{s^2 = 100}\)

\(\mathrm{s = 10}\)

Answer: D) 10

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may confuse volume formulas for different 3D shapes. They might use \(\mathrm{V = s^3}\) (thinking it's a cube) or \(\mathrm{V = \frac{1}{3}\pi r^2h}\) (confusing with a cone), leading to incorrect setups and wrong calculations.

This leads to confusion and incorrect answer selection.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{300 = 3s^2}\) but make algebraic errors, such as getting \(\mathrm{s^2 = 900}\) instead of \(\mathrm{s^2 = 100}\), or forgetting to take the square root properly.

This may lead them to select Choice E) 17 or cause calculation confusion.

The Bottom Line:

This problem requires recognizing the specific volume formula for a square pyramid and carefully executing the algebra. The key insight is that knowing two of the three variables (volume, height, base area) allows you to solve for the third.