Question

The length of the minute hand for a clock is 7 cm. Find the area swept by it in 40 minutes.

Hint:

Remember the speeds of the clock hands: the minute hand moves 6\(^\circ\) per minute, and the hour hand moves 0.5\(^\circ\) per minute. These are useful for various clock-related problems.

Answer 1

Step 1: Understanding the Concept:
The area swept by the minute hand of a clock forms a sector of a circle. To find this area, we need to determine the central angle of the sector created in the given time and use the formula for the area of a sector. The length of the minute hand will be the radius of this circle.

Step 2: Key Formula or Approach:
1. Find the angle swept by the minute hand. The minute hand completes a full circle (360\(^\circ\)) in 60 minutes.
Angle swept in 1 minute = \(\frac{360^\circ}{60} = 6^\circ\).
2. The formula for the area of a sector is: \[ \text{Area} = \frac{\theta}{360^\circ} \times \pi r^2 \] where \(\theta\) is the central angle and \(r\) is the radius.

Step 3: Detailed Explanation:
Given:
Radius (length of minute hand), \(r = 7\) cm.
Time = 40 minutes.
First, calculate the angle \(\theta\) swept in 40 minutes: \[ \theta = 40 \text{ minutes} \times 6^\circ/\text{minute} = 240^\circ \] Now, calculate the area of the sector using the formula with \(\pi = \frac{22}{7}\): \[ \text{Area} = \frac{240}{360} \times \frac{22}{7} \times (7)^2 \] Simplify the fraction: \[ \frac{240}{360} = \frac{24}{36} = \frac{2}{3} \] Substitute this back into the area formula: \[ \text{Area} = \frac{2}{3} \times \frac{22}{7} \times 49 \] \[ \text{Area} = \frac{2}{3} \times 22 \times 7 \] \[ \text{Area} = \frac{308}{3} \text{ cm}^2 \] This can also be written as \(102.67\) cm\(^2\).

Step 4: Final Answer:
The area swept by the minute hand in 40 minutes is \(\frac{308}{3}\) cm\(^2\).