Points to Remember:
- The area covered by the minute hand represents a sector of a circle.
- The formula for the area of a sector is (θ/360) * Ïr², where θ is the central angle in degrees and r is the radius.
- The minute hand completes a full circle (360°) in 60 minutes.
Introduction:
This question requires a factual and analytical approach to determine the area swept by the minute hand of a clock in a given time. The problem involves calculating the area of a sector of a circle. The minute hand, with a length of 14 cm, acts as the radius of the circle. We need to determine the angle swept by the minute hand in 5 minutes and then use this angle to calculate the area of the sector.
Body:
1. Calculating the Angle:
The minute hand completes a full rotation (360°) in 60 minutes. Therefore, in 5 minutes, it covers (5/60) * 360° = 30°.
2. Calculating the Area of the Sector:
The area of a sector is given by the formula: Area = (θ/360) * Ïr²
Where:
- θ = central angle (in degrees) = 30°
- r = radius = length of the minute hand = 14 cm
- Ï â 3.14159
Substituting the values:
Area = (30/360) * Ï * (14)²
Area = (1/12) * Ï * 196
Area â (1/12) * 3.14159 * 196
Area â 51.31 cm²
Therefore, the area covered by the minute hand in 5 minutes is approximately 51.31 square centimeters.
Conclusion:
In summary, by applying the formula for the area of a sector of a circle and considering the proportion of a full rotation covered by the minute hand in 5 minutes, we calculated the area covered to be approximately 51.31 cm². This problem demonstrates a practical application of geometry in everyday scenarios. No policy recommendations or best practices are applicable in this purely mathematical context. The solution highlights the importance of understanding fundamental geometric principles and their application in solving real-world problems. The accuracy of the calculation depends on the precision of the value of Ï used. For more precise results, a more accurate value of Ï should be employed.