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: Understand the problem:
We are given that the minute hand of a clock is 21 cm long, and we are asked to find the area swept by it in 10 minutes.

Step 2: Concept of the area swept by the minute hand:
The area swept by the minute hand of the clock is the sector of a circle with the minute hand as the radius. The area of a sector of a circle is given by the formula:
\[ \text{Area of the sector} = \frac{\theta}{360^\circ} \times \pi r^2 \] where \( \theta \) is the angle subtended by the sector, and \( r \) is the radius of the circle (which is the length of the minute hand).

Step 3: Find the angle subtended by the minute hand in 10 minutes:
The minute hand of the clock completes one full rotation (360°) in 60 minutes. Therefore, the angle subtended by the minute hand in 1 minute is:
\[ \frac{360^\circ}{60} = 6^\circ \] In 10 minutes, the minute hand will sweep an angle of:
\[ \theta = 10 \times 6^\circ = 60^\circ \]

Step 4: Calculate the area of the sector:
Now that we know the angle \( \theta = 60^\circ \) and the radius \( r = 21 \) cm, we can calculate the area swept by the minute hand in 10 minutes:
\[ \text{Area of the sector} = \frac{60^\circ}{360^\circ} \times \pi (21)^2 \] Simplify the fraction \( \frac{60^\circ}{360^\circ} = \frac{1}{6} \), so the area becomes:
\[ \text{Area of the sector} = \frac{1}{6} \times \pi \times 21^2 = \frac{1}{6} \times \pi \times 441 = \frac{441\pi}{6} = 73.5\pi \, \text{sq. cm} \]

Step 5: Conclusion:
The area swept by the minute hand in 10 minutes is \( \boxed{73.5\pi \, \text{sq. cm}} \). If you approximate \( \pi \approx 3.1416 \), then the area is approximately:
\[ 73.5 \times 3.1416 = 231 \, \text{sq. cm} \] So, the area swept by the minute hand is approximately \( 231 \, \text{sq. cm} \).