A wall clock has a minute hand of length 7 cm. Find the area swept by the minute hand in 30 minutes (i.e., in a 180° rotation).

Introduction / Context:
The tip of a clock’s minute hand traces a circle of radius equal to the hand’s length. The “area swept” in a given time is the sector area corresponding to the angular turn in that time. In 30 minutes, the minute hand turns through 180 degrees (half a revolution), so the swept area is simply half the area of the full circle formed by the minute hand’s rotation.

Given Data / Assumptions:

• Radius r = 7 cm.
• Angular turn in 30 minutes = 180 degrees = π radians.
• Circle area formula: A = πr^2.
• Sector area (θ in radians): A_sector = (θ / (2π)) * πr^2 = (θ/2) * r^2.

Concept / Approach:
Because 30 minutes is exactly half a turn for the minute hand, the area swept is one-half of the full circle’s area. This avoids any need for degree-to-radian conversion beyond recognizing that 180 degrees is half a circle.

Step-by-Step Solution:
Full circle area = π * r^2 = π * 7^2 = 49πSwept fraction in 30 minutes = 1/2 of full revolutionArea swept = (1/2) * 49π = 24.5πUsing π = 22/7, area = 24.5 * 22/7 = 77 sq. cm

Verification / Alternative check:
Compute via sector formula directly with θ = π radians: A_sector = (θ/2) * r^2 = (π/2) * 49 = 24.5π ⇒ 77 with π = 22/7, agreeing with the above.

Why Other Options Are Wrong:
147 sq. cm is the full circle area (49π) rounded with π ≈ 3; 210 and 154 sq. cm correspond to mismatched multiples of π for different radii, not for 7 cm and a half-turn.

Common Pitfalls:
Forgetting that 30 minutes is half a rotation; using diameter instead of radius; or taking half the circumference instead of half the area.

Final Answer:
77 sq. cm