A right square pyramid has a volume of 3{,}072 cubic centimeters and a base area of 576 square centimeters. What...

1. TRANSLATE the problem information

• Given information:
  • Volume = 3,072 cubic centimeters
  • Base area = 576 square centimeters
  • Need to find slant height (apex to midpoint of base edge)

2. INFER the solution strategy

• To find slant height, we need to think about the geometry:
  • Slant height connects apex to midpoint of base edge
  • This creates a right triangle with the pyramid height
  • We need the pyramid height and the horizontal distance first
• Start by finding the pyramid height using the volume formula

3. SIMPLIFY to find the pyramid height

• Use pyramid volume formula: \(\mathrm{V = \frac{1}{3}Bh}\)
• Substitute: \(\mathrm{3{,}072 = \frac{1}{3}(576)h}\)
• Simplify: \(\mathrm{3{,}072 = 192h}\)
• Solve: \(\mathrm{h = 3{,}072 \div 192 = 16}\) cm

4. INFER what we need for the Pythagorean theorem

• The slant height forms a right triangle where:
  • Vertical leg = pyramid height = 16 cm
  • Horizontal leg = distance from base center to midpoint of edge
  • We need to find that horizontal distance

5. SIMPLIFY to find the base dimensions

• Base area = 576, so side length: \(\mathrm{s^2 = 576}\)
• Therefore: \(\mathrm{s = \sqrt{576} = 24}\) cm
• Distance from center to midpoint of edge = \(\mathrm{s/2 = 12}\) cm

6. SIMPLIFY using Pythagorean theorem

• slant height = \(\mathrm{\sqrt{height^2 + horizontal\:distance^2}}\)
• slant height = \(\mathrm{\sqrt{16^2 + 12^2}}\)
  \(\mathrm{= \sqrt{256 + 144}}\)
  \(\mathrm{= \sqrt{400}}\)
  \(\mathrm{= 20}\) cm

Answer: B. 20

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the geometric relationship for slant height and try to use the volume or base area directly in some incorrect formula.

They might attempt to use the slant height in the volume formula or create an incorrect relationship like "slant height = volume ÷ base area." This leads to confusion and random guessing among the answer choices.

Second Most Common Error:

Conceptual confusion about geometric setup: Students find the height correctly (h = 16) but then use the wrong horizontal distance in the Pythagorean theorem.

Instead of using s/2 = 12 (center to midpoint of edge), they might use the full side length s = 24 (center to corner). This gives \(\mathrm{\sqrt{16^2 + 24^2}}\)
\(\mathrm{= \sqrt{256 + 576}}\)
\(\mathrm{= \sqrt{832}}\)
\(\mathrm{\approx 28.8}\), leading them to select Choice D. 29.

The Bottom Line:

This problem requires students to work through a multi-step process: volume → height → geometric visualization → Pythagorean theorem. The key challenge is INFERRING the correct geometric setup for the right triangle that contains the slant height.