A hemisphere has radius 21 cm. Using π = 22/7, what is the volume of this hemisphere in cubic centimetres?

Introduction / Context:
This question tests knowledge of the volume formula for a hemisphere. Understanding volumes of standard solid shapes such as spheres, hemispheres, cylinders, and cones is essential in aptitude tests and engineering entrance examinations.

Given Data / Assumptions:
- The solid is a hemisphere, which is half of a sphere.
- Radius r = 21 cm.
- Use π = 22/7 as specified.
- We must find the volume in cubic centimetres.

Concept / Approach:
The volume of a sphere of radius r is given by:
Volume of sphere = (4 / 3) * π * r^3.
A hemisphere is exactly half of a sphere, so its volume is:
Volume of hemisphere = (1 / 2) * (4 / 3) * π * r^3 = (2 / 3) * π * r^3.
We substitute r = 21 cm and π = 22/7 into this formula and evaluate.

Step-by-Step Solution:
Step 1: Write the formula for the volume of a hemisphere: V = (2 / 3) * π * r^3. Step 2: Substitute r = 21 cm and π = 22/7. Step 3: Compute r^3 = 21^3 = 21 * 21 * 21 = 9261. Step 4: Substitute into the formula: V = (2 / 3) * (22 / 7) * 9261. Step 5: First simplify 9261 / 7 = 1323, so V = (2 / 3) * 22 * 1323. Step 6: Compute 22 * 1323 = 29106. Step 7: Now multiply by 2 / 3: V = (2 / 3) * 29106 = 58212 / 3 = 19404 cubic centimetres.

Verification / Alternative Check:
As a quick approximate check, using π close to 3.14, the volume would be about (2 / 3) * 3.14 * 21^3. This also leads to a value near 19400 cubic centimetres, supporting that 19404 cubic centimetres is correct when using π = 22/7 exactly.

Why Other Options Are Wrong:
Option a, 38808 cubic cms, is exactly double the correct answer and corresponds to the volume of a full sphere rather than a hemisphere.
Option b, 9702 cubic cms, is half the correct volume and might result from an incorrect division or misapplied factor.
Option d, 29106 cubic cms, is an intermediate product obtained before multiplying by 2 / 3, so it represents a partial calculation rather than the final answer.

Common Pitfalls:
Students often forget that a hemisphere is half of a sphere and directly use the formula (4 / 3) * π * r^3, which yields the volume of the full sphere. Another common mistake is mishandling the fractions when simplifying, especially the division by 7 and multiplication by 2 / 3. Writing each step clearly and simplifying systematically prevents such errors.

Final Answer:
The volume of the hemisphere is 19404 cubic cms.