Calculating the Radius of a Wheel from Revolutions and Distance Traversed

In this article, we will delve into the mathematical relationship between the radius of a wheel, the number of revolutions it makes, and the distance it covers. Understanding this concept is essential for various real-world applications, such as mechanical engineering, transportation, and physics.

Understanding the Relationship Between Distance, Revolutions, and Circumference

The circumference of a wheel is key to understanding how far the wheel travels in one full revolution. The relationship is straightforward: the distance covered in one revolution is equal to the circumference of the wheel. To find the radius, we use the formula for the circumference of a circle, which is (2pi r). Here, (r) represents the radius of the wheel and (pi) is approximately 3.14159.

The First Example: 500 Revolutions Covering 48 km

Problem Statement: A wheel makes 500 revolutions in covering a distance of 48 km. What is the radius of the wheel?

Solution:

Convert the total distance from kilometers to meters. Since 1 km 1000 m, the total distance in meters is: Calculate the circumference of the wheel using the distance and number of revolutions: Use the circumference to find the radius of the wheel: Convert 48 km to meters: 48 km 48000 m Calculate the circumference:text{Circumference} frac{48000 text{ m}}{500} 96 text{ m} Rearrange the formula for the circumference to solve for the radius:r frac{96 text{ m}}{2pi} approx frac{96}{6.2832} approx 15.29 text{ m}

Thus, the radius of the wheel is approximately 15.29 meters.

The Second Example: 4000 Revolutions Covering 60 km

Problem Statement: A wheel makes 4000 revolutions covering a distance of 60 km. What is the radius of the wheel?

Solution:

Calculate the distance covered in one revolution: Determine the circumference of the wheel using the radius: Convert 60 km to meters:60 km 601000 m Calculate the distance covered in one revolution:1 revolution 15 m Rearrange the circumference formula to solve for the radius:2 pi r 15 r frac{15}{2pi} approx 2.386363approx 2.39 m

Thus, the radius of the wheel is approximately 2.39 meters.

The Third Example: 1200 Revolutions Covering 60 km

Problem Statement: If a wheel covers a distance of 60 km by making 1200 revolutions, what is the radius of the wheel?

Solution:

Find the distance covered in one revolution using the circumference formula: Use the circumference to find the radius: Convert 60 km to meters:60 km 60000 m Distance covered in 1200 revolutions:2400 pi r 60000 r frac{60000}{2400 pi} frac{25}{pi} Substitute the value of (pi) and calculate the radius: r frac{25}{3.14159} approx 7.95 m

Thus, the radius of the wheel is approximately 7.95 meters.

Conclusion

By understanding the relationship between the radius, circumference, and the number of revolutions, you can accurately determine the size of a wheel based on the distance it travels. This knowledge is crucial for various applications, including the design and maintenance of vehicles and machinery.