The total surface area of a hemisphere is 462 square centimetres. Using π = 22/7, what is the curved surface area of this hemisphere in square centimetres?

Introduction / Context:
This question tests understanding of the relationship between total surface area and curved surface area for a hemisphere. Such problems are very common in mensuration topics in aptitude and school examinations.

Given Data / Assumptions:
- The solid is a hemisphere.
- Total surface area (TSA) = 462 sq cm.
- π = 22/7 is to be used.
- We need the curved surface area (CSA) of the hemisphere.

Concept / Approach:
For a hemisphere of radius r, the following formulas hold:
- Curved surface area (CSA) = 2 * π * r^2.
- Total surface area (TSA) = curved surface area + area of the circular base = 2 * π * r^2 + π * r^2 = 3 * π * r^2.
Given TSA, we can first find r by solving 3 * π * r^2 = 462, then compute CSA = 2 * π * r^2.

Step-by-Step Solution:
Step 1: Use the TSA formula: 3 * π * r^2 = 462. Step 2: Substitute π = 22/7 to get 3 * (22/7) * r^2 = 462. Step 3: Simplify the constant factor: 3 * 22 / 7 = 66 / 7. Step 4: The equation becomes (66 / 7) * r^2 = 462. Step 5: Multiply both sides by 7: 66 * r^2 = 462 * 7 = 3234. Step 6: Divide by 66: r^2 = 3234 / 66 = 49. Step 7: Therefore r = 7 cm. Step 8: Now compute CSA = 2 * π * r^2 = 2 * (22/7) * 7^2. Step 9: Evaluate: 7^2 = 49, so CSA = 2 * (22/7) * 49 = 2 * 22 * 7 = 308 sq cm.

Verification / Alternative Check:
We can verify by recomputing TSA using r = 7 cm: TSA = 3 * π * r^2 = 3 * (22/7) * 49 = 3 * 22 * 7 = 462 sq cm, which matches the given total surface area. Hence the radius and curved surface area are consistent with the original information.

Why Other Options Are Wrong:
Option a, 616 sq cms, is equal to 4 * π * r^2 for r = 7 and would correspond to the surface area of a full sphere, not a hemisphere.
Option b, 154 sq cms, is only half of the base area when r = 7 and has no direct meaning in this context.
Option d, 462 sq cms, is the total surface area, not just the curved part, so it includes the circular base as well.

Common Pitfalls:
Students sometimes confuse TSA and CSA for hemispheres and may use TSA = 2 * π * r^2 instead of 3 * π * r^2. Others attempt to guess the radius instead of solving the equation systematically. Remembering that for hemispheres TSA = 3 * π * r^2 and CSA = 2 * π * r^2 helps to quickly set up the correct equations.

Final Answer:
The curved surface area of the hemisphere is 308 sq cms.