The minutes of hand a clock is `7` cm long. The area traced out by the minutes hand of clock between `4:15` pm to `4:35` pm on a day is

Options

A. `5(1)/(3)cm^(2)`
B. `51(1)/(3)cm^(2)`
C. `(152)/(3)cm^(2)`
D. `(151)/(3)cm^(2)`

Answer

To find the area traced out by the minute hand of a clock between 4:15 PM and 4:35 PM, we can follow these steps:

### Step 1: Understand the movement of the minute hand
The minute hand moves around the clock face, which is a circle. The minute hand is 7 cm long, which is the radius of the circle traced by the minute hand.

### Step 2: Calculate the time interval
From 4:15 PM to 4:35 PM, the time interval is:
\[ 4:35 - 4:15 = 20 \text{ minutes} \]

### Step 3: Find the area of the entire circle
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
Where \( r \) is the radius. Here, \( r = 7 \) cm.

Substituting the value of \( r \):
\[ A = \pi (7)^2 = \pi \times 49 \]

Using \( \pi \approx \frac{22}{7} \):
\[ A = \frac{22}{7} \times 49 = 154 \text{ cm}^2 \]

### Step 4: Calculate the area covered in 20 minutes
Since the minute hand completes a full circle (360 degrees) in 60 minutes, we need to find the area covered in 20 minutes. The area covered in 20 minutes is a fraction of the total area based on the time ratio:
\[ \text{Area covered in 20 minutes} = \frac{20}{60} \times 154 \]
This simplifies to:
\[ \text{Area covered in 20 minutes} = \frac{1}{3} \times 154 = \frac{154}{3} \text{ cm}^2 \]

### Step 5: Convert to mixed number
To express \( \frac{154}{3} \) as a mixed number:
- Divide 154 by 3:
\[ 154 \div 3 = 51 \text{ remainder } 1 \]
So, \( \frac{154}{3} = 51 \frac{1}{3} \text{ cm}^2 \).

### Final Answer
The area traced out by the minute hand of the clock between 4:15 PM and 4:35 PM is:
\[ 51 \frac{1}{3} \text{ cm}^2 \]