The length of the minute hand of clock is 14 cm. Find the area swept by the minute hand in 15 minutes.

Answer

To find the area swept by the minute hand of a clock in 15 minutes, we can follow these steps:

### Step 1: Understand the movement of the minute hand
The minute hand completes a full circle (360 degrees) in 60 minutes.

### Step 2: Calculate the angle covered in 1 minute
To find the angle covered in 1 minute, we divide the total degrees in a circle by the number of minutes in an hour:
\[ \text{Angle covered in 1 minute} = \frac{360 \text{ degrees}}{60 \text{ minutes}} = 6 \text{ degrees} \]

### Step 3: Calculate the angle covered in 15 minutes
Now, we can find the angle covered in 15 minutes:
\[ \text{Angle covered in 15 minutes} = 15 \text{ minutes} \times 6 \text{ degrees/minute} = 90 \text{ degrees} \]

### Step 4: Use the area formula for a sector
The area \(A\) of a sector of a circle can be calculated using the formula:
\[ A = \frac{\pi r^2 \theta}{360} \]
where \(r\) is the radius and \(\theta\) is the angle in degrees.

### Step 5: Substitute the values into the formula
Given that the length of the minute hand (the radius \(r\)) is 14 cm and the angle \(\theta\) is 90 degrees:
\[ A = \frac{\pi \times (14)^2 \times 90}{360} \]

### Step 6: Simplify the expression
First, calculate \(14^2\):
\[ 14^2 = 196 \]
Now substitute this value into the area formula:
\[ A = \frac{\pi \times 196 \times 90}{360} \]
Next, simplify \(\frac{90}{360}\):
\[ \frac{90}{360} = \frac{1}{4} \]
So, we can rewrite the area as:
\[ A = \frac{\pi \times 196}{4} \]

### Step 7: Calculate the area
Now, calculate \(\frac{196}{4}\):
\[ \frac{196}{4} = 49 \]
Thus, the area becomes:
\[ A = 49\pi \]
Using \(\pi \approx \frac{22}{7}\):
\[ A = 49 \times \frac{22}{7} = \frac{1078}{7} \approx 154 \text{ cm}^2 \]

### Final Answer
Therefore, the area swept by the minute hand in 15 minutes is approximately:
\[ \text{Area} \approx 154 \text{ cm}^2 \]