A right pyramid has a square base. The side length of the square base is 13/3 meters, and the height...

1. TRANSLATE the problem information

• Given information:
  • Right pyramid with square base
  • Side length of square base: \(\frac{13}{3}\) meters
  • Height of pyramid: \(\frac{7}{2}\) meters
  • Need to find: Volume in cubic meters

• What this tells us: We have all the measurements needed to apply the pyramid volume formula.

2. INFER the approach

• Since we have a pyramid, we'll use: \(\mathrm{V} = \frac{1}{3} \times \mathrm{Base\:Area} \times \mathrm{Height}\)
• First we need the base area, then substitute everything into the volume formula
• The base is square, so we'll need to square the side length

3. Calculate the base area

• Base Area = \(\left(\mathrm{side\:length}\right)^2 = \left(\frac{13}{3}\right)^2\)
• Base Area = \(\frac{169}{9}\) square meters

4. SIMPLIFY by applying the volume formula

• \(\mathrm{V} = \frac{1}{3} \times \frac{169}{9} \times \frac{7}{2}\)
• Multiply the numerators: \(1 \times 169 \times 7 = 1183\)
• Multiply the denominators: \(3 \times 9 \times 2 = 54\)
• \(\mathrm{V} = \frac{1183}{54}\) cubic meters

Answer: B. \(\frac{1183}{54}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when multiplying the fractions, particularly when calculating \(1 \times 169 \times 7 = 1183\) or \(3 \times 9 \times 2 = 54\). Common mistakes include getting 1173 instead of 1183 in the numerator, or miscalculating the denominator as 36 or 27.

This may lead them to select Choice A (\(\frac{1183}{36}\)) or Choice C (\(\frac{1183}{27}\)).

Second Most Common Error:

Missing conceptual knowledge about pyramid volume: Students might confuse the pyramid volume formula with other volume formulas, forgetting the \(\frac{1}{3}\) factor and just multiplying Base Area × Height directly.

This leads to getting \(\frac{1183}{18}\) and selecting Choice D.

The Bottom Line:

This problem tests careful fraction arithmetic more than complex geometric reasoning. The key is systematically working through the multiplication without rushing the calculations.