Find the volume, in cubic centimetres, of a solid hemisphere whose diameter is 21 cm. Take π = 22/7 for the calculation.

Introduction / Context:
This problem focuses on the volume of a hemisphere, which is a common three dimensional solid in mensuration and aptitude questions. A hemisphere is exactly half of a sphere, so its volume can be obtained by applying the standard sphere volume formula and then halving it. Competitive exams often test your ability to correctly identify which formula to use, substitute the given values accurately, and handle fractions and constants like π efficiently. Here you are given the diameter of the hemisphere and a specific approximate value for π, and you are asked to compute the volume in cubic centimetres.

Given Data / Assumptions:
• The solid is a hemisphere (half of a sphere).
• Diameter of the hemisphere = 21 cm.
• Therefore, radius r = 21 / 2 = 10.5 cm.
• Use π = 22 / 7 as an approximation for calculations.
• We need the volume expressed in cubic centimetres (cubic cm).

Concept / Approach:
The volume of a full sphere of radius r is given by V_sphere = (4 / 3) * π * r^3. A hemisphere is half of a sphere, so its volume is V_hemisphere = (1 / 2) * V_sphere = (2 / 3) * π * r^3. The key is to convert the diameter into the radius first, then substitute into the formula, and finally perform the arithmetic carefully. Using π = 22 / 7 helps produce a rational numerical result that matches the given options exactly.

Step-by-Step Solution:
Step 1: Compute the radius from the given diameter: r = 21 / 2 = 10.5 cm. Step 2: Recall the volume formula for a hemisphere: V = (2 / 3) * π * r^3. Step 3: Substitute π = 22 / 7 and r = 10.5 cm into the formula. Step 4: First compute r^3: 10.5 * 10.5 * 10.5 = 1157.625 cubic cm. Step 5: Now compute V = (2 / 3) * (22 / 7) * 1157.625. Step 6: Perform the multiplication and division stepwise; the simplified exact result is 2425.5 cubic cm.

Verification / Alternative check:
To verify, you can approximate using π ≈ 3.14 and see if the result is close to the chosen option. With π ≈ 3.14 and r ≈ 10.5, the value of (2 / 3) * π * r^3 will also be around 2400 to 2500 cubic cm, which confirms that 2425.5 cubic cm is reasonable. Additionally, volumes of spheres and hemispheres grow with the cube of the radius, so a radius slightly above 10 cm would indeed yield a volume in the low thousands of cubic centimetres, which supports the magnitude of the answer.

Why Other Options Are Wrong:
The option 2235.5 cubic cm is somewhat smaller and can come from an arithmetic slip, such as incorrectly computing r^3 or forgetting part of the fraction (2 / 3). The options 2040 and 1860 cubic cm are significantly lower and would result from either using π incorrectly (for example, approximating it as 3) or mistakenly using the formula for a cylinder or another shape. Only 2425.5 cubic cm matches the correct substitution into the hemisphere volume formula with π = 22 / 7 and radius 10.5 cm.

Common Pitfalls:
Common mistakes include using the diameter instead of the radius directly in the formula, forgetting that the volume is for a hemisphere and not a full sphere, or misapplying the formula for the volume of a cylinder or cone. Another source of error is rushing through the computation of r^3 and making multiplication errors. To avoid these pitfalls, always convert from diameter to radius first, double check that you are using V = (2 / 3) * π * r^3 for a hemisphere, and perform the arithmetic in clear steps, preferably with intermediate checks or estimation to ensure the result is in a sensible range.

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