A right rectangular prism A has a square base and a volume of 80 cubic centimeters. Prism B also has...

1. TRANSLATE the problem information

• Given information:
  • Prism A: square base, volume = 80 cm³
  • Prism B: square base with side = \(\mathrm{\frac{3}{2}}\) × side of A, height = \(\mathrm{\frac{1}{2}}\) × height of A
• Let s = side length of A's base, h = height of A
• Then: \(\mathrm{s^2h = 80}\)

2. INFER how dimensions affect volume

• Key insight: When dimensions scale, volume scales by the product of all scaling factors
• For prism B:
  • Base side becomes \(\mathrm{\frac{3}{2}s}\), so base area becomes \(\mathrm{[\frac{3}{2}s]^2 = \frac{9}{4}s^2}\)
  • Height becomes \(\mathrm{\frac{1}{2}h}\)
• Therefore: \(\mathrm{Volume\;of\;B = \frac{9}{4}s^2 \times \frac{1}{2}h = \frac{9}{8}s^2h}\)

3. SIMPLIFY to find the final answer

• Since \(\mathrm{s^2h = 80}\) for prism A:
  \(\mathrm{Volume\;of\;B = \frac{9}{8} \times 80}\)
  \(\mathrm{= \frac{9 \times 80}{8}}\)
  \(\mathrm{= \frac{720}{8}}\)
  \(\mathrm{= 90}\)

Answer: D (90)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students forget that when the base side length scales by 3/2, the base area scales by \(\mathrm{(\frac{3}{2})^2 = \frac{9}{4}}\), not just 3/2.

They calculate:
\(\mathrm{Volume = \frac{3}{2} \times \frac{1}{2} \times 80}\)
\(\mathrm{= \frac{3}{4} \times 80}\)
\(\mathrm{= 60}\)

This leads them to select Choice B (60).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the scaling factor as 9/8 but make arithmetic errors.

Some might calculate \(\mathrm{\frac{9}{8} \times 80}\) incorrectly, perhaps getting \(\mathrm{9 \times 10 = 90}\) but then second-guessing themselves and picking a different answer, or making errors like treating 9/8 as 1.125 and getting confused with decimal calculations.

The Bottom Line:

This problem tests understanding of how scaling in multiple dimensions affects volume - it's not just about multiplying individual scaling factors, but recognizing that area scales by the square of linear scaling.