A roller of diameter 70 cm and length 2 m is rolling on the ground. What is the area covered by the roller in 50 revolutions ?

Answer

To find the area covered by a roller in 50 revolutions, we can follow these steps:

### Step 1: Identify the dimensions of the roller
- The diameter of the roller is given as 70 cm.
- The length (or height) of the roller is given as 2 m.

### Step 2: Convert the diameter to radius
- The radius \( r \) is half of the diameter.

\[ r = \frac{\text{Diameter}}{2} = \frac{70 \text{ cm}}{2} = 35 \text{ cm} \]

### Step 3: Convert radius to meters
- Since the length is given in meters, we should convert the radius from centimeters to meters.

\[ r = \frac{35 \text{ cm}}{100} = 0.35 \text{ m} \]

### Step 4: Use the formula for the curved surface area (CSA) of the cylinder
- The formula for the curved surface area of a cylinder is:

\[ \text{CSA} = 2 \pi r h \]

- Here, \( h \) is the length of the roller, which is 2 m.

### Step 5: Substitute the values into the formula
- Plug in the values into the CSA formula:

\[ \text{CSA} = 2 \times \pi \times 0.35 \text{ m} \times 2 \text{ m} \]

### Step 6: Calculate the CSA
- Using \( \pi \approx \frac{22}{7} \):

\[ \text{CSA} = 2 \times \frac{22}{7} \times 0.35 \times 2 \]

### Step 7: Simplify the calculation
- First, calculate \( 2 \times 0.35 \):

\[ 2 \times 0.35 = 0.7 \]

- Now substitute back:

\[ \text{CSA} = 2 \times \frac{22}{7} \times 0.7 \]

- Multiply \( 2 \times 0.7 = 1.4 \):

\[ \text{CSA} = \frac{22 \times 1.4}{7} \]

- Calculate \( 22 \times 1.4 = 30.8 \):

\[ \text{CSA} = \frac{30.8}{7} \approx 4.4 \text{ m}^2 \]

### Step 8: Calculate the area covered in 50 revolutions
- The area covered by the roller in 50 revolutions is:

\[ \text{Total Area} = 50 \times \text{CSA} = 50 \times 4.4 \text{ m}^2 \]

### Step 9: Final calculation
- Calculate \( 50 \times 4.4 \):

\[ \text{Total Area} = 220 \text{ m}^2 \]

### Final Answer
The area covered by the roller in 50 revolutions is **220 m²**.

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