Difficulty: Medium
Correct Answer: 30%
Explanation:
Introduction / Context:This question compares the numerical values of a sphere’s surface area and volume for a given radius. Although area and volume have different units, aptitude problems often request the percentage ratio of their numerical magnitudes for a fixed measurement.
Given Data / Assumptions:Radius r = 10 cm. Surface area S = 4πr^2. Volume V = (4/3)πr^3.
Concept / Approach:Compute S/V * 100%. The π factors cancel. The power of r reduces to 1 in the denominator, leaving a simple arithmetic fraction multiplied by 100. Substitute r = 10 to obtain a clean percentage.
Step-by-Step Solution:
S = 4πr^2 = 4π * 100 = 400π V = (4/3)πr^3 = (4/3)π * 1000 = 4000π/3 S/V = (400π) / (4000π/3) = (400 * 3) / 4000 = 1200/4000 = 0.3 Percentage = 0.3 * 100 = 30%Verification / Alternative check:Approximate numerically: S ≈ 1256.6; V ≈ 4188.8; S/V ≈ 0.3 ⇒ 30%.
Why Other Options Are Wrong:Values like 24%, 26.5%, or 45% do not match the exact ratio. 33.3% is a common distractor (one-third) but incorrect here.
Common Pitfalls:Forgetting the factor 1/3 in volume, or failing to cancel π. Misplacing powers of r can also lead to wrong ratios.
Final Answer:30%
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