The surface area of a sphere is 5544 square centimetres. Using the standard formula for the surface area of a sphere, what is the diameter of the sphere (in cm)?

Introduction / Context:
This question is from the topic of mensuration and deals with the surface area of a sphere. We are given the total surface area and asked to find the diameter. To solve it, we use the standard formula for the surface area of a sphere in terms of its radius and then convert the radius into the diameter. This is a direct formula substitution question commonly seen in aptitude and school level exams.

Given Data / Assumptions:
- Surface area S of the sphere is 5544 square centimetres. - Surface area formula for a sphere: S = 4 * π * r^2. - π is taken as 22 / 7 unless specified otherwise in typical questions of this type. - We must find the diameter, which is 2 * r.

Concept / Approach:
The main idea is to express the unknown radius r in terms of the known surface area S using the formula S = 4 * π * r^2. Once we isolate r^2, we take the square root to find r, and then multiply by 2 to get the diameter. Keeping the arithmetic clear and handling π as 22 / 7 helps in quickly simplifying to an integer answer.

Step-by-Step Solution:
Step 1: Write the formula for the surface area of a sphere: S = 4 * π * r^2. Step 2: Substitute S = 5544 and π = 22 / 7 into the formula. So 5544 = 4 * (22 / 7) * r^2. Step 3: Simplify the constant part: 4 * (22 / 7) = 88 / 7. Step 4: The equation becomes 5544 = (88 / 7) * r^2. Step 5: Multiply both sides by 7 to clear the denominator: 5544 * 7 = 88 * r^2. Step 6: Compute 5544 * 7 = 38808. So 38808 = 88 * r^2. Step 7: Divide both sides by 88 to get r^2 = 38808 / 88. Step 8: Simplify 38808 / 88. Cancelling by 8 gives 4851 / 11, and with full simplification, r^2 = 441. Step 9: Take the square root: r = √441 = 21 cm. Step 10: Diameter of the sphere is 2 * r = 2 * 21 = 42 cm.

Verification / Alternative check:
To verify, substitute r = 21 cm back into the formula S = 4 * π * r^2. We get S = 4 * (22 / 7) * 21^2 = 4 * (22 / 7) * 441. Simplify (441 / 7) = 63, so S = 4 * 22 * 63 = 88 * 63 = 5544 square centimetres. This matches the given surface area, confirming that r = 21 cm and diameter = 42 cm is correct.

Why Other Options Are Wrong:
Option 21 is the radius, not the diameter. The question specifically asks for the diameter. Option 84 would correspond to a radius of 42 cm, which would give a much larger surface area than 5544 square centimetres. Option 63 would correspond to a radius of 31.5 cm, again inconsistent with the given surface area.

Common Pitfalls:
One common mistake is to stop once the radius is found and forget to double it to obtain the diameter. Another is to mishandle π and the fractions during simplification, especially when multiplying and dividing by 7. To avoid mistakes, perform each simplification step carefully and check by substituting back into the original formula whenever possible.

Final Answer:
The diameter of the sphere is 42 cm.