Two similar right circular cylinders A and B have total surface areas of 72 square centimeters and 288 square centimeters,...

1. TRANSLATE the problem information

• Given information:
  • Cylinder A: Surface area = \(\mathrm{72\ cm^2}\)
  • Cylinder B: Surface area = \(\mathrm{288\ cm^2}\), Volume = \(\mathrm{576\ cm^3}\)
  • The cylinders are similar (corresponding dimensions are proportional)
• We need to find: Sum of both volumes

2. INFER the scaling approach

• Since the cylinders are similar, we can use scaling relationships
• Key insight: We have surface areas for both cylinders, which will help us find the scale factor
• Strategy: Find scale factor from surface area ratio → use volume scaling → find volume of A → add volumes

3. SIMPLIFY to find the surface area ratio

• Surface area ratio = \(\mathrm{\frac{288}{72} = 4}\)
• This means cylinder B has 4 times the surface area of cylinder A

4. INFER the linear scale factor

• For similar figures, surface area scales by \(\mathrm{k^2}\) where k is the linear scale factor
• Since surface area ratio is 4: \(\mathrm{k^2 = 4}\)
• Therefore: \(\mathrm{k = 2}\)
• This means cylinder B's linear dimensions are 2 times those of cylinder A

5. INFER and SIMPLIFY the volume relationship

• For similar figures, volume scales by \(\mathrm{k^3}\)
• Volume ratio = \(\mathrm{k^3 = 2^3 = 8}\)
• This means cylinder B has 8 times the volume of cylinder A

6. SIMPLIFY to find volume of cylinder A

• Volume of A = Volume of B ÷ 8 = 576 ÷ 8 = \(\mathrm{72\ cm^3}\)

7. SIMPLIFY to find the final answer

• Sum = Volume of A + Volume of B = 72 + 576 = \(\mathrm{648\ cm^3}\)

Answer: 648

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse the scaling relationships and apply the wrong power to the scale factor. They might think volume scales by \(\mathrm{k^2}\) instead of \(\mathrm{k^3}\), or use the surface area ratio directly as the volume ratio.

For example, they might incorrectly calculate: Volume of A = 576 ÷ 4 = \(\mathrm{144\ cm^3}\), leading to a sum of 144 + 576 = \(\mathrm{720\ cm^3}\). This leads to confusion since 720 isn't typically among answer choices, causing them to guess.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret which cylinder is larger, thinking A is the larger cylinder because it's mentioned first. They might try to find volume of B from A instead of the other way around.

This backward setup leads to incorrect calculations and eventually to abandoning systematic solution and guessing.

The Bottom Line:

This problem requires students to connect the abstract concept of similarity to concrete scaling relationships. The key challenge is recognizing that you need to work backwards from the given volume of the larger cylinder to find the volume of the smaller one.