Wire circle reshaped to rectangle: A circular wire has radius 42 cm. It is cut and bent into a rectangle whose sides are in the ratio 6 : 5. Find the smaller side of the rectangle.

Introduction / Context:
Perimeter is conserved when a wire is reshaped. A circle turned into a rectangle retains its total length; with the side ratio given, we can recover the actual sides.

Given Data / Assumptions:

• Circle radius r = 42 cm
• Rectangle sides ratio = 6 : 5 → sides 6x and 5x
• Perimeter(rectangle) = Circumference(circle)

Concept / Approach:
Circumference C = 2 * π * r. Rectangle perimeter P = 2(6x + 5x) = 22x. Equate C and P to solve x, then compute the smaller side 5x.

Step-by-Step Solution:
C = 2 * π * 42 = 84π cmWith π = 22/7, C = 84 * 22/7 = 264 cm22x = 264 → x = 12 cmSmaller side = 5x = 60 cm

Verification / Alternative check:
Larger side = 6x = 72 cm; perimeter = 2(72 + 60) = 264 cm which matches the original circumference, confirming correctness.

Why Other Options Are Wrong:
30 cm and 36 cm misuse the ratio; 72 cm is the larger side; 132 cm is the full half-perimeter, not a side.

Common Pitfalls:
Setting 6x + 5x equal to circumference instead of 2(6x + 5x); rounding π unnecessarily when a neat 22/7 result exists.

Final Answer:
60 cm