Calculate Road Roller Revolutions for Area Coverage

The problem asks for the number of revolutions a road roller makes to cover a specific area. We need to find the area covered in one revolution and divide the total area by this value.

Unit Conversion

First, ensure all units are consistent. Convert the given measurements from centimeters (cm) to meters (m).

• Road roller width (h): 80 cm = 0.8 m
• Road roller radius (r): 77 cm = 0.77 m
• Total area to cover (A): $96.8 \text{ m}^2$

Area Covered Per Revolution

In one revolution, the road roller covers an area equal to its lateral surface area (like the label of a cylinder). The formula for the lateral surface area (LSA) of a cylinder is $A_{rev} = 2 \pi r h$.

Using the given values and $\pi = \frac{22}{7}$:

$ A_{rev} = 2 \times \frac{22}{7} \times 0.77 \text{ m} \times 0.8 \text{ m} $

Calculate the value:

$ A_{rev} = 2 \times \frac{22}{7} \times \frac{77}{100} \times \frac{8}{10} \text{ m}^2 $

$ A_{rev} = 2 \times 22 \times \frac{11}{100} \times \frac{8}{10} \text{ m}^2 \quad (\text{since } \frac{77}{7} = 11) $

$ A_{rev} = \frac{44 \times 11 \times 8}{1000} \text{ m}^2 $

$ A_{rev} = \frac{3872}{1000} \text{ m}^2 $

$ A_{rev} = 3.872 \text{ m}^2 $

So, the roller covers $3.872 \text{ m}^2$ in one revolution.

Calculate Number of Revolutions

To find the total number of revolutions (N), divide the total area to be covered by the area covered in one revolution.

$ N = \frac{\text{Total Area}}{\text{Area per Revolution}} $

$ N = \frac{96.8 \text{ m}^2}{3.872 \text{ m}^2} $

Perform the division:

$ N = \frac{96.8}{3.872} = \frac{96800}{3872} $

Simplifying the fraction:

$ N = \frac{12100}{484} = \frac{3025}{121} = 25 $

Therefore, the roller will take 25 revolutions.