Two sculptures, Sculpture A and Sculpture B, are similar right rectangular prisms. The height of Sculpture B is 1/3 the...

1. TRANSLATE the problem information

• Given information:
  • Sculptures A and B are similar right rectangular prisms
  • Height of B = \(\frac{1}{3}\) × Height of A
  • Surface area of A = 540 square feet
  • Find surface area of B

• What this tells us: The scale factor from A to B is \(\frac{1}{3}\)

2. INFER the scaling relationship

• Key insight: For similar figures, surface area doesn't scale the same way as linear dimensions
• Linear dimensions scale by the scale factor \(\frac{1}{3}\)
• Surface area scales by the square of the scale factor: \((\frac{1}{3})^2 = \frac{1}{9}\)

3. SIMPLIFY to find the answer

• Surface area of B = Surface area of A × (scale factor)²

\(\mathrm{Surface\;area\;of\;B} = \mathrm{Surface\;area\;of\;A} \times (\mathrm{scale\;factor})^2\)

\(\mathrm{Surface\;area\;of\;B} = 540 \times \frac{1}{9} = 60\)

Answer: 60

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students use the linear scale factor directly for surface area instead of squaring it.

They reason: "If B's height is \(\frac{1}{3}\) of A's height, then B's surface area should be \(\frac{1}{3}\) of A's surface area." This leads to calculating \(540 \times \frac{1}{3} = 180\).

This may lead them to select Choice (C) (180).

Second Most Common Error:

Inadequate SIMPLIFY execution: Students understand the concept but make calculation errors with fractions.

They might incorrectly calculate \((\frac{1}{3})^2\) as \(\frac{1}{6}\) instead of \(\frac{1}{9}\), or struggle with \(540 \div 9\), leading to wrong numerical results.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests whether students truly understand how scaling affects different measurements. The key insight is that area measurements (2-dimensional) scale as the square of linear measurements (1-dimensional), not at the same rate.