For a right circular cylinder, let A be the sum of the total surface area and the area of the two circular bases, and let B be the curved surface area. If A : B = 3 : 2 and the volume of the cylinder is 4312 cm3, what is the sum of the areas of the two bases in square centimetres? Take pi = 22/7.

Introduction / Context:
This problem links ratios of surface areas to the volume of a right circular cylinder. It requires careful interpretation of what A and B represent, followed by algebraic manipulation of the ratio conditions. Once the relation between curved surface area and base area is found, the volume information can be used to determine the radius and finally the combined area of the two circular bases. This chain of reasoning is a classic example of mensuration based ratio questions.

Given Data / Assumptions:

• A is defined as total surface area plus the area of the two bases.
• B is defined as the curved surface area.
• For the cylinder, A : B = 3 : 2.
• Volume of the cylinder = 4312 cm3.
• Right circular cylinder with radius r and height h.
• Use pi = 22/7 in calculations.

Concept / Approach:
For a cylinder with radius r and height h: curved surface area C = 2 * pi * r * h, base area each = pi * r^2, and total surface area T = C + 2 * pi * r^2. The problem defines A = T + 2 * pi * r^2 = C + 4 * pi * r^2, and B = C. From A : B = 3 : 2, an equation in terms of C and base area is obtained. This yields a relationship between C and the base area. Then the volume V = pi * r^2 * h is used to find numerical values of r and h that satisfy this relation and the given volume.

Step-by-Step Solution:
Step 1: Let C = curved surface area = 2 * pi * r * h, and let B0 = area of one base = pi * r^2.Step 2: Total surface area T = C + 2B0. Given A = T + 2B0 = C + 4B0.Step 3: The ratio A : B is (C + 4B0) : C = 3 : 2.Step 4: Write (C + 4B0) / C = 3 / 2. This simplifies to 1 + 4B0 / C = 3 / 2. Hence 4B0 / C = 1/2, so C = 8B0.Step 5: Substitute C = 2 * pi * r * h and B0 = pi * r^2 to get 2 * pi * r * h = 8 * pi * r^2.Step 6: Cancel pi * r on both sides to obtain 2h = 8r, so h = 4r.Step 7: Volume V = pi * r^2 * h = pi * r^2 * 4r = 4 * pi * r^3 = 4312.Step 8: Using pi = 22/7, 4 * (22/7) * r^3 = 4312 gives (88/7) * r^3 = 4312. Thus r^3 = (4312 * 7) / 88 = 343, so r = 7 cm.Step 9: Base area B0 = pi * r^2 = (22/7) * 49 = 154 cm2. Sum of areas of two bases = 2B0 = 308 cm2.

Verification / Alternative check:
Using r = 7 and h = 28 (since h = 4r), volume V = pi * 7^2 * 28 = (22/7) * 49 * 28 = 4312 cm3, which matches the given volume. The curved surface area C = 2 * pi * r * h = 2 * (22/7) * 7 * 28 = 1232 cm2. Base area each is 154 cm2, so T = C + 2B0 = 1232 + 308 = 1540 cm2. A = T + 2B0 = 1540 + 308 = 1848 cm2. Ratio A : B = 1848 : 1232 simplifies to 3 : 2, confirming the algebra and the radius used, and hence confirming that 308 cm2 for the two bases is correct.

Why Other Options Are Wrong:

• 154 cm2 is the area of one base, not the sum of both bases.
• 462 cm2 and 616 cm2 would require different values of r or h that fail the given volume and ratio conditions.
• 231 cm2 is neither one base nor two bases for any r satisfying the required relations.

Common Pitfalls:

• Misinterpreting A as only total surface area instead of total surface area plus two base areas.
• Forgetting to simplify the ratio A : B correctly and missing the key relation C = 8B0.
• Making arithmetic mistakes when solving for r^3 and r.

Final Answer:
The sum of the areas of the two bases of the cylinder is 308 cm2.