Difficulty: Easy
Correct Answer: 14 : 11
Explanation:
Introduction:
This question compares the areas of two shapes, a square and a circle, when they share a common linear dimension: the side of the square equals the diameter of the circle. It tests your ability to manipulate formulas and form an exact ratio of areas.
Given Data / Assumptions:
Concept / Approach:
Area of a square with side s is s². Area of a circle with radius r is πr². By substituting r = s/2 and simplifying, we can express both areas in terms of s and π. Then we can form their ratio and get a fraction independent of s.
Step-by-Step Solution:
Area of square Aₛ = s². Radius of circle r = s/2. Area of circle A𝚌 = πr² = π(s/2)² = π(s² / 4) = (π/4)s². We need Aₛ : A𝚌 = s² : (π/4)s². Cancel s² (since s ≠ 0): ratio = 1 : (π/4) = 4 : π. Now use π = 22/7, so ratio = 4 : (22/7). Multiply both terms by 7 to clear denominator: 4 * 7 : 22 = 28 : 22. Simplify 28 : 22 by dividing both parts by 2: 14 : 11.
Verification / Alternative check:
If you choose a specific value, say s = 14 units, then square area is 14² = 196. Circle radius r = 7, area = π * 7² = (22/7) * 49 = 154. Ratio 196 : 154 simplifies to 14 : 11, confirming the general derivation.
Why Other Options Are Wrong:
Ratios 28 : 11, 7 : 22, and 22 : 7 either ignore the step of simplification or invert the ratio. Only 14 : 11 maintains the exact proportional relationship consistent with both the formulas and the numerical check.
Common Pitfalls:
A common error is to forget that the circle's radius is half the diameter, leading to area proportional to s² instead of (s/2)². Others may accidentally invert the ratio and give area(circle) : area(square). Being explicit with formulas avoids these mistakes.
Final Answer:
The ratio of the area of the square to the area of the circle is 14 : 11.
Discussion & Comments