Two solid spheres are geometrically similar. The volume of the larger sphere is 640 cubic centimeters and the volume of...

1. TRANSLATE the problem information

• Given information:
  • Larger sphere: \(\mathrm{V = 640\text{ cm}^3}\), \(\mathrm{r = 18\text{ cm}}\)
  • Smaller sphere: \(\mathrm{V = 80\text{ cm}^3}\), \(\mathrm{r = ?}\)
  • The spheres are geometrically similar
• What this tells us: We need to use the scaling properties of similar solids to connect the volumes and radii.

2. INFER the mathematical relationship

• For similar 3D solids, volumes scale as the cube of linear dimensions
• If \(\mathrm{k = \frac{\text{radius of smaller}}{\text{radius of larger}}}\), then:
  \(\mathrm{\frac{\text{Volume of smaller}}{\text{Volume of larger}} = k^3}\)
• This is different from 2D figures where areas scale as \(\mathrm{k^2}\)

3. TRANSLATE to set up the equation

• Volume ratio = \(\mathrm{\frac{80}{640} = \frac{1}{8}}\)
• So we have: \(\mathrm{k^3 = \frac{1}{8}}\)

4. SIMPLIFY to find the linear scaling factor

• Take the cube root of both sides: \(\mathrm{k = \sqrt[3]{\frac{1}{8}}}\)
• Since \(\mathrm{\sqrt[3]{\frac{1}{8}} = \frac{\sqrt[3]{1}}{\sqrt[3]{8}} = \frac{1}{2}}\), we get \(\mathrm{k = \frac{1}{2}}\)
• This means the smaller radius is \(\mathrm{\frac{1}{2}}\) times the larger radius

5. SIMPLIFY to find the final answer

• \(\mathrm{\text{radius}_{\text{small}} = k \times \text{radius}_{\text{large}} = \frac{1}{2} \times 18 = 9\text{ cm}}\)

Answer: (D) 9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse 2D and 3D scaling relationships and use \(\mathrm{k^2}\) instead of \(\mathrm{k^3}\) for the volume ratio.

They set up: \(\mathrm{k^2 = \frac{80}{640} = \frac{1}{8}}\), so \(\mathrm{k = \sqrt{\frac{1}{8}} = \frac{1}{2\sqrt{2}} \approx 0.35}\), giving \(\mathrm{\text{radius} \approx 0.35 \times 18 \approx 6.4}\). This may lead them to select Choice (C) (6) as the closest answer.

Second Most Common Error:

Poor TRANSLATE reasoning: Students set up the ratio backwards, thinking larger/smaller instead of smaller/larger.

They calculate: \(\mathrm{k^3 = \frac{640}{80} = 8}\), so \(\mathrm{k = 2}\), giving \(\mathrm{\text{radius} = 2 \times 18 = 36}\). Since 36 isn't an option, this leads to confusion and guessing.

The Bottom Line:

The key insight is recognizing that 3D similar solids have volume ratios equal to the cube (not square) of their linear dimension ratios. Students who successfully identify this relationship can solve the problem systematically, while those who confuse 2D and 3D scaling rules get trapped in incorrect calculations.