Question

The minute hand of a clock is 21 cm long. The area swept by it in 10 minutes is :

• A. $121 \, \text{cm}^2$
• B. $131 \, \text{cm}^2$
• C. $231 \, \text{cm}^2$
• D. $172.5 \, \text{cm}^2$

Answer 1

The Correct Option is C

Step 1: Problem Definition
Determine the area traced by a 21 cm minute hand on a clock in 10 minutes. This area corresponds to a sector of a circle.

Step 2: Sector Area Formula
The area \(A\) of a circular sector with radius \(r\) and central angle \(\theta\) (in radians) is: \[A = \frac{1}{2} r^2 \theta\] Here, \(r = 21 \, \text{cm}\).

Step 3: Calculate Central Angle
A minute hand completes a full circle ( \(2\pi\) radians) in 60 minutes.
For 10 minutes, the fraction of the revolution is: \[\frac{10}{60} = \frac{1}{6}\] The corresponding central angle is: \[\theta = \frac{1}{6} \times 2\pi = \frac{\pi}{3} \, \text{radians}\]

Step 4: Compute Swept Area
Using \(r = 21 \, \text{cm}\) and \(\theta = \frac{\pi}{3}\) radians in the sector area formula: \[A = \frac{1}{2} \times (21)^2 \times \frac{\pi}{3}\] Calculation: \[A = \frac{1}{2} \times 441 \times \frac{\pi}{3} = \frac{441\pi}{6} = 73.5\pi \, \text{cm}^2\] Approximate value using \(\pi \approx 3.1416\): \[A \approx 73.5 \times 3.1416 = 231 \, \text{cm}^2\]

Step 5: Final Result
The area swept by the minute hand in 10 minutes is approximately \(231 \, \text{cm}^2\).