Two right circular cylinders have volumes in the ratio 7 : 3 and heights in the ratio 7 : 9. If the area of the base of the second cylinder is 154 sq cm, what is the radius (in cm) of the first cylinder?

Introduction / Context:
This problem tests the relation between volume, base area and height of right circular cylinders. When ratios of volumes and heights are known, we can deduce the ratio of base areas, and then find an unknown radius using the standard area formula for a circle. This type of question checks comfort with proportional reasoning and formulas for circular area and cylinder volume in aptitude and competitive exams.

Given Data / Assumptions:

• Ratio of volumes of two cylinders V1 : V2 = 7 : 3.
• Ratio of their heights h1 : h2 = 7 : 9.
• Area of base of the second cylinder A2 = 154 sq cm.
• Both cylinders are right circular cylinders.
• Use the usual formula volume = base area * height.

Concept / Approach:
For a right circular cylinder, volume V = A * h, where A is the base area and h is the height. If we know the ratio of volumes and heights, we can obtain the ratio of base areas using V1 / V2 = (A1 * h1) / (A2 * h2). After finding A1, we use the circle area formula A1 = π * r1^2 to obtain the radius of the first cylinder. The numbers are chosen so that the radius can be written neatly in surd form involving √3.

Step-by-Step Solution:
Step 1: Write the volume ratio in terms of base areas and heights: V1 / V2 = (A1 * h1) / (A2 * h2). Step 2: Substitute the given ratios: 7 / 3 = (A1 / A2) * (7 / 9). Step 3: Rearrange to get A1 / A2 = (7 / 3) * (9 / 7) = 9 / 3 = 3. Step 4: Therefore A1 = 3 * A2 = 3 * 154 = 462 sq cm. Step 5: Use A1 = π * r1^2 with π taken as 22 / 7, so r1^2 = A1 * 7 / 22 = 462 * 7 / 22 = 147. Step 6: Simplify r1^2 = 147 = 49 * 3, hence r1 = 7√3 cm.

Verification / Alternative check:
We can quickly verify A2. With π = 22 / 7, A2 = 154 sq cm corresponds to r2^2 = 154 * 7 / 22 = 49, so r2 = 7 cm, which is a neat radius. The ratio A1 / A2 = 462 / 154 = 3 as expected. Using r1 = 7√3, the base area of the first cylinder is π * (7√3)^2 = π * 147 = 462 sq cm, matching A1. Hence the computed radius is consistent with all given ratios, so the answer is reliable.

Why Other Options Are Wrong:
6√2 and 6√3 give r1^2 values that do not match 147 when substituted into π * r1^2. The option 7√2 gives r1^2 = 98, which would give base area smaller than 462 sq cm. The extra option 5√3 also leads to a base area that does not match the required multiple of 154 sq cm. Only 7√3 produces the correct base area and satisfies the given volume and height ratios.

Common Pitfalls:
A common mistake is to directly compare the radii using the volume ratio without accounting for the different heights. Another error is to manipulate the fraction (7 / 3) * (7 / 9) instead of dividing by (7 / 9). Some learners also forget to use π consistently, which may lead to arithmetic mismatches. Carefully applying the relation V proportional to A * h and simplifying ratios step by step avoids these problems.

Final Answer:
The radius of the first cylinder is 7√3 cm.