Difficulty: Easy
Correct Answer: 4140π sq. cm
Explanation:
Introduction:
This problem involves the area of an annulus, which is the region between two concentric circles (circles with the same centre but different radii). The key idea is that the area of this ring is simply the difference between the areas of the larger and the smaller circles.
Given Data / Assumptions:
Concept / Approach:
The area of a circle is given by Area = πr². For two concentric circles, the annular area (ring area) is simply the difference of their areas. So, Annular area = πR² − πr² = π(R² − r²). We will compute R² and r², subtract, and then multiply by π.
Step-by-Step Solution:
Compute the squares:R² = 68² = 4624.r² = 22² = 484.Difference R² − r² = 4624 − 484 = 4140.Therefore, area of the annulus = π × 4140 = 4140π sq. cm.
Verification / Alternative check:
You can separately compute the two circle areas: larger circle = 4624π, smaller circle = 484π, and then subtract: 4624π − 484π = 4140π, which matches the earlier calculation. Both procedures lead to the same result, confirming that the computation is correct.
Why Other Options Are Wrong:
4110π, 4080π, 4050π, and 4000π sq. cm correspond to incorrect differences of the squared radii. They would result from wrong squaring, incorrect subtraction, or mixing up one of the radii. Only 4140π sq. cm is consistent with the formula π(R² − r²) using R = 68 cm and r = 22 cm.
Common Pitfalls:
Common mistakes include subtracting the radii first and then squaring (which is incorrect), or forgetting to square one of the radii. Always remember that areas depend on the square of the radius, so you must compute R² and r² before subtracting. Another mistake is to try to plug in π approximately even though the question clearly asks for an answer in terms of π.
Final Answer:
The area of the region between the two concentric circles is 4140π sq. cm.
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