Right rectangular prism X is similar to right rectangular prism Y. The surface area of right rectangular prism X is...

1. TRANSLATE the problem information

• Given information:
  • Prisms X and Y are similar
  • Surface area of X = \(\mathrm{58\text{ cm}^2}\)
  • Surface area of Y = \(\mathrm{1{,}450\text{ cm}^2}\)
  • Volume of Y = \(\mathrm{1{,}250\text{ cm}^3}\)
• Need to find: sum of volumes of both prisms

2. INFER the scaling relationships

• Since the prisms are similar, all linear dimensions of Y are k times those of X (where k is the scale factor)
• This means:
  • Surface areas are related by \(\mathrm{k^2}\)
  • Volumes are related by \(\mathrm{k^3}\)
• Strategy: Use the surface area ratio to find k, then use \(\mathrm{k^3}\) to find volume of X

3. SIMPLIFY to find the scale factor

• Surface area of Y = \(\mathrm{k^2}\) × Surface area of X
• \(\mathrm{1{,}450 = k^2 \times 58}\)
• \(\mathrm{k^2 = 1{,}450 \div 58 = 25}\) (use calculator)
• \(\mathrm{k = \sqrt{25} = 5}\)

4. SIMPLIFY to find volume of X

• Volume of Y = \(\mathrm{k^3}\) × Volume of X
• \(\mathrm{1{,}250 = 5^3 \times Volume\text{ }of\text{ }X}\)
• \(\mathrm{1{,}250 = 125 \times Volume\text{ }of\text{ }X}\)
• Volume of X = \(\mathrm{1{,}250 \div 125 = 10\text{ cm}^3}\) (use calculator)

5. Calculate final answer

• Sum of volumes = Volume of X + Volume of Y
• Sum = \(\mathrm{10 + 1{,}250 = 1{,}260\text{ cm}^3}\)

Answer: 1,260

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the scaling relationships for similar 3D figures. They might try to directly compare the given values without understanding that surface areas scale by \(\mathrm{k^2}\) and volumes by \(\mathrm{k^3}\). This leads to confusion and random attempts at division or multiplication without systematic reasoning. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the scaling relationships but make calculation errors. Common mistakes include getting \(\mathrm{k^2 = 25}\) but forgetting to take the square root (using \(\mathrm{k = 25}\) instead of \(\mathrm{k = 5}\)), or incorrectly calculating \(\mathrm{k^3 = 5^3 = 25}\) instead of 125. This may lead them to calculate an incorrect volume for prism X and get a wrong final sum.

The Bottom Line:

This problem requires students to bridge conceptual understanding of similarity with algebraic manipulation. The key insight is recognizing that different properties of similar figures scale differently—surface area by \(\mathrm{k^2}\) and volume by \(\mathrm{k^3}\)—then systematically using these relationships.