91Ó°ÊÓ

In the context of circles, the radius is a very important measure. It is the distance from the center of the circle to any point on its edge. You can think of the radius like the spoke of a wheel, stretching from the central hub to the tire's edge. This measure not only helps us understand the size of the circle but is also crucial in calculating other properties, like the circumference.

The formula for circumference, which is the distance around the circle, relies heavily on the radius:

• Circumference formula: \( C = 2\pi r \)
• Where \( r \) stands for the radius.

In our example, the wheel's radius is 2 feet. So, when applying the formula, the circumference becomes \( 4\pi \) feet. By understanding the radius, you grasp the first essential step in determining the overall distance a wheel will travel.

Revolutions may sound complex, but they simply represent how many times a wheel turns entirely around its axis. Imagine spinning in one place; one complete spin equates to one revolution. For a wheel, each revolution coils through one full circumference of the circle.

So, the number of revolutions indicates the number of times the wheel travels its own circumference. This helps us calculate longer distances:

• One revolution moves the distance of one circumference.
• Multiply the wheel's circumference by the number of revolutions to find the total travel distance.

In our exercise, the wheel makes 150 revolutions. With each revolution covering \( 4\pi \) feet, multiple these to calculate the entire journey. Understanding revolutions allows us to connect one full turn of a wheel to distance traveled.

When it comes to finding out how far the wheel has rolled, you multiply the circumference by the revolution count. This method uses both the wheel's size, captured by the circumference formula, and the frequency of its full turns, contained in the revolutions count.

Here’s the step-by-step process:

• Calculate the circumference using \( C = 2\pi r \) where \( r = 2 \text{ feet} \).
• Resulting in \( 4\pi \) feet per revolution.
• Multiply the circumference \( 4\pi \) feet by 150 revolutions.
• Total distance traveled: \( 600\pi \) feet.

This calculation demonstrates how understanding each component separately gives you the total distance accurately. This approach helps you quantify long distance travel based on simple, repeatable cycles of turning.