The diameter of a solid hemisphere is 21 cm. What is the volume of the hemisphere in cubic centimetres?

Introduction / Context:
This mensuration question focuses on the volume of a hemisphere. Many aptitude and technical exams include questions where a hemisphere is used directly or as part of another solid, so knowing the formula and how to substitute values correctly is important.

Given Data / Assumptions:
• The solid in question is a hemisphere. • Diameter of the hemisphere is 21 cm. • We are asked to find its volume in cubic centimetres.

Concept / Approach:
The volume of a full sphere of radius r is V_sphere = (4 / 3) * π * r^3. A hemisphere is exactly half of a sphere, so its volume is V_hemisphere = (1 / 2) * V_sphere = (2 / 3) * π * r^3. First we compute the radius r from the diameter, then apply this formula. When approximate numeric answers are given, π is often taken as 22 / 7.

Step-by-Step Solution:
1. The diameter D of the hemisphere is 21 cm. 2. The radius r is half the diameter: r = D / 2 = 21 / 2 = 10.5 cm. 3. Volume of a hemisphere is V = (2 / 3) * π * r^3. 4. Compute r^3: r = 10.5, so r^3 = 10.5 * 10.5 * 10.5. 5. First 10.5 * 10.5 = 110.25. Then 110.25 * 10.5 = 1157.625. 6. So r^3 = 1157.625. 7. Substitute into the volume formula: V = (2 / 3) * π * 1157.625. 8. If we use π = 22 / 7, then V = (2 / 3) * (22 / 7) * 1157.625. 9. Compute (2 / 3) * (22 / 7) ≈ 44 / 21. 10. Multiply 44 / 21 by 1157.625. The exact product simplifies to approximately 2425.5 cubic centimetres.

Verification / Alternative check:
We can approximate using π ≈ 3.14 as well. Then V ≈ (2 / 3) * 3.14 * 1157.625. Compute (2 / 3) * 3.14 ≈ 2.0933 and then multiply by 1157.625 to get a value close to 2424.5 cubic centimetres. This is very near 2425.5, confirming that 2425.5 cubic centimetres is an accurate rounded answer when using π = 22 / 7.

Why Other Options Are Wrong:
• 2810 cub.cm: This is larger than what is obtained from the correct formula and overestimates the volume. • 1250.5 cub.cm: This is roughly half the correct volume and would correspond to using incorrect dimensions or formula. • 1725.25 cub.cm: This lies between half and three quarters of the correct value and has no basis in the standard formula. • 2100 cub.cm: This might be guessed as 100 * 21 but does not come from the hemisphere volume formula.

Common Pitfalls:
Students sometimes forget to divide the diameter by 2 to get the radius, or they mistakenly use the full sphere volume formula without halving it. Another common error is miscomputing r^3 or approximating π too crudely. Always write the formula, substitute values carefully and then evaluate step by step.

Final Answer:
The volume of the hemisphere is 2425.5 cub.cm.