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: Understanding the problem:
We are asked to find the area swept by the minute hand of a clock in 10 minutes. The minute hand is 21 cm long.
The minute hand traces a circular arc, and the area swept by the minute hand in 10 minutes is the area of a sector of the circle.

Step 2: Formula for the area of a sector:
The area \(A\) of a sector of a circle with radius \(r\) and central angle \(\theta\) (in radians) is given by the formula: \[ A = \frac{1}{2} r^2 \theta \] where \(r = 21 \, \text{cm}\) is the radius of the circle, and \(\theta\) is the central angle in radians.

Step 3: Finding the central angle (\(\theta\)) for 10 minutes:
The minute hand completes one full revolution (360° or \(2\pi\) radians) in 60 minutes.
In 10 minutes, the minute hand sweeps a fraction of the full revolution. The fraction is: \[ \frac{10}{60} = \frac{1}{6} \] Thus, the central angle \(\theta\) for 10 minutes is: \[ \theta = \frac{1}{6} \times 2\pi = \frac{\pi}{3} \, \text{radians} \]

Step 4: Calculating the area swept:
Now that we know the central angle \(\theta = \frac{\pi}{3}\) radians and the radius \(r = 21 \, \text{cm}\), we can substitute these values into the formula for the area of the sector: \[ A = \frac{1}{2} \times (21)^2 \times \frac{\pi}{3} \] Simplifying: \[ A = \frac{1}{2} \times 441 \times \frac{\pi}{3} = \frac{441\pi}{6} = 73.5\pi \, \text{cm}^2 \] Approximating \(\pi \approx 3.1416\): \[ A \approx 73.5 \times 3.1416 = 231 \, \text{cm}^2 \]

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