Two concentric circles (same centre) are drawn. The circumference of the smaller circle is 44 cm, and the circumference of the larger circle is double that of the smaller circle. What is the area (in square cm) of the region lying between these two circles?

Introduction / Context:
This question is about concentric circles and the area of the ring shaped region (annulus) between them. It tests understanding of the relationship between circumference and radius, as well as the formula for the area of a circle.

Given Data / Assumptions:

• The circumference of the smaller circle is 44 cm.
• The circumference of the larger circle is 88 cm, which is double 44 cm.
• Both circles share the same centre (they are concentric).
• We can use π = 22 / 7 for calculations.
• We need the area of the region between the two circles, that is π(R^2 − r^2), where R is the larger radius and r is the smaller radius.

Concept / Approach:
For any circle, the circumference C is related to its radius r by: C = 2πr From each given circumference we can compute the corresponding radius. Then we find the area of each circle using: Area = πr^2 Finally, the area between the two circles (the annulus) is: Area between = π(R^2 − r^2)

Step-by-Step Solution:
Step 1: For the smaller circle, C1 = 44 cm. Step 2: Use 44 = 2πr1. With π = 22 / 7, we have 2π = 44 / 7. Step 3: So r1 = 44 / (44 / 7) = 7 cm. Step 4: For the larger circle, C2 = 88 cm. Step 5: Again use C2 = 2πR, so R = 88 / (44 / 7) = 14 cm. Step 6: Now compute R^2 − r^2 = 14^2 − 7^2 = 196 − 49 = 147. Step 7: Area between the circles = π(R^2 − r^2) = π * 147. Step 8: Substitute π = 22 / 7 to get area = (22 / 7) * 147 = 22 * 21 = 462 square cm.

Verification / Alternative check:
We can also compute the full areas separately. Area of larger circle = πR^2 = (22 / 7) * 196 = 22 * 28 = 616 square cm. Area of smaller circle = πr^2 = (22 / 7) * 49 = 22 * 7 = 154 square cm. Difference = 616 − 154 = 462 square cm, which matches our direct computation.

Why Other Options Are Wrong:
154 square cm: This is the area of the smaller circle alone, not the annular region.
308 square cm: This is neither the difference nor the sum of the two areas and does not match any correct calculation.
616 square cm: This is the area of the larger circle alone.
210 square cm: This value is not derived from any consistent use of circumference and area formulas.

Common Pitfalls:
A frequent mistake is to average the radii or circumferences instead of computing areas using the correct formulas. Some students also forget to subtract the smaller area from the larger one. Another common error is mishandling the constant π or using an inconsistent approximation. Always follow the sequence: circumference to radius, then radius to area, and finally area difference.

Final Answer:
The area of the region between the two circles is 462 square cm.