Two concentric circles (circles with the same centre) have radii 68 cm and 22 cm respectively. What is the area, in terms of π, of the closed region (ring-shaped area) bounded between the two circular boundaries?

Introduction / Context:
This question focuses on the area of a ring-shaped region (annulus) formed by two concentric circles. The region of interest is the area between the outer circle and the inner circle. To find this, we subtract the area of the smaller inner circle from the area of the larger outer circle. This is a very standard geometry problem that reinforces understanding of the circle area formula and basic algebraic subtraction.

Given Data / Assumptions:

• Radius of the outer circle, R = 68 cm.
• Radius of the inner circle, r = 22 cm.
• Both circles share the same centre (they are concentric).
• We need the area of the region between the two circles in square centimetres, in terms of π.

Concept / Approach:
The area of a circle with radius r is π * r^2. For an annulus formed by two concentric circles, the area of the ring is equal to the area of the larger circle minus the area of the smaller circle. So we use the formula: area of annulus = π * R^2 − π * r^2 = π * (R^2 − r^2). The problem is then reduced to computing R^2 and r^2, subtracting them, and multiplying the result by π.

Step-by-Step Solution:
Outer radius R = 68 cm, inner radius r = 22 cm.Area of outer circle = π * R^2 = π * 68^2.Area of inner circle = π * r^2 = π * 22^2.Compute 68^2: 68 * 68 = 4624.Compute 22^2: 22 * 22 = 484.Difference R^2 − r^2 = 4624 − 484 = 4140.Area of annulus = π * (R^2 − r^2) = π * 4140.Thus the required area = 4140π square centimetres.

Verification / Alternative check:
We can quickly verify the arithmetic. The inner area is much smaller because r is significantly less than R. The outer circle area is 4624π and the inner circle area is 484π, so the difference is (4624 − 484)π = 4140π, confirming the subtraction is correct. The difference 4624 − 4000 = 624 and 624 − 484 = 140, so the calculation step by step is consistent. This check reassures us that the numeric difference is accurate.

Why Other Options Are Wrong:
Options like 4110π, 4080π, 4050π, and 4000π all result from incorrect subtraction or miscalculated squares of 68 or 22. For example, if someone mistakenly used 67 or 21 instead, or miscalculated 68^2, they might arrive at one of those wrong values. Only 4140π sq.cm. corresponds to the correct computation of π * (68^2 − 22^2).

Common Pitfalls:
A common mistake is to add the areas of the two circles instead of subtracting, forgetting that the inner circle region is removed from the ring. Another frequent error is miscalculating the squares of 68 or 22. Sometimes students forget to keep π as a common factor and instead approximate it early, which can lead to rounding errors. Keeping π symbolically and focusing on accurate integer arithmetic avoids these problems.

Final Answer:
The area of the ring-shaped region between the two concentric circles is 4140π sq.cm.