The diameter of a roller, 1 m long, is 84 cm. If it takes 500 complete revolutions to level a playground, the area of the playground is

Options

A. `1440 m^(2)`
B. `1320 m^(2)`
C. `1260 m^(2)`
D. `1550 m^(2)`

Explanation

To find the area of the playground that a roller levels, we can follow these steps:

### Step 1: Convert the dimensions to the same unit
The diameter of the roller is given as 1 meter and 84 cm. We need to convert everything to centimeters for consistency.
- 1 meter = 100 cm
- Therefore, the diameter of the roller is 100 cm.

### Step 2: Calculate the radius of the roller
The radius \( R \) is half of the diameter.
\[ R = \frac{\text{Diameter}}{2} = \frac{100 \text{ cm}}{2} = 50 \text{ cm} \]

### Step 3: Identify the height of the roller
The height \( H \) of the roller is given as 1 meter, which is:
\[ H = 100 \text{ cm} \]

### Step 4: Calculate the curved surface area of the roller
The formula for the curved surface area (CSA) of a cylinder is given by:
\[ \text{CSA} = 2 \pi R H \]
Substituting the values:
\[ \text{CSA} = 2 \times \frac{22}{7} \times 50 \text{ cm} \times 100 \text{ cm} \]
Calculating this step by step:
\[ \text{CSA} = 2 \times \frac{22}{7} \times 50 \times 100 \]
\[ = \frac{44}{7} \times 5000 \]
\[ = \frac{220000}{7} \approx 31428.57 \text{ cm}^2 \]

### Step 5: Calculate the total area covered after 500 revolutions
To find the total area covered by the roller after 500 revolutions, we multiply the curved surface area by the number of revolutions:
\[ \text{Total Area} = \text{CSA} \times \text{Number of Revolutions} \]
\[ = 31428.57 \text{ cm}^2 \times 500 \]
\[ = 15714285 \text{ cm}^2 \]

### Step 6: Convert the area from cm² to m²
To convert square centimeters to square meters, we divide by 10,000:
\[ \text{Area in m}^2 = \frac{15714285 \text{ cm}^2}{10000} = 1571.4285 \text{ m}^2 \]

### Final Answer
The area of the playground is approximately:
\[ \text{Area} \approx 1571.43 \text{ m}^2 \]