Problem 65 - Prealgebra

Question

The wheel on a 26 -inch bicycle is such that the distance from the center of the wheel to the outside of the tire is 26.75 inches. If you walk the bicycle so that the wheel turns through one complete revolution, how many inches did you walk? Round to the nearest inch.

Step-by-Step Solution

Answer

The bicycle travels approximately 168 inches in one full revolution of the wheel.

Step 1: Identify the Problem

We need to find how far the bicycle travels when its wheel makes one full revolution. This requires knowing the circumference of the wheel, which is the distance the wheel travels in one revolution.

Step 2: Determine the Radius of the Wheel

The distance from the center of the wheel to the outside, the radius, is given as 26.75 inches. Thus, the radius \( r \) is 26.75 inches.

Step 3: Apply the Formula for Circumference

The formula for circumference \( C \) is \( C = 2 \pi r \). We will use this formula to calculate the wheel's circumference.

Step 4: Calculate the Circumference

Substitute \( r = 26.75 \) into the formula: \[ C = 2 \pi (26.75) \approx 2 \cdot 3.1416 \cdot 26.75 = 168.1193 \] inches.

Step 5: Round the Result

Finally, round \( 168.1193 \) to the nearest whole number, which gives \( 168 \) inches.

Key Concepts

Understanding Geometry in a Circle

Geometry plays a crucial role in calculating the circumference of a circle. When you think about a circular wheel, you realize it's a symmetrical shape where every point on the edge is exactly the same distance from the center. This distance is what we call the radius. The complete journey around this edge is known as the circumference. In geometry, the circle is defined by two important measurements: the radius and diameter, which are interconnected. The diameter is twice the radius and stretches from one side of the circle to the other, passing through the center. Therefore, knowing either the diameter or radius is fundamental for further geometric calculations. They're not just numbers, but key pieces in the puzzle of understanding the circle's properties. So, any task involving circles, like finding out how far a wheel travels when it makes a full revolution, demands a solid grasp of these geometric principles.

Exploring Radius and Diameter

In the realm of circles, both the radius and diameter are incredibly important. Let's break these terms down for better understanding.

• The radius is the distance from the center of the circle to any point on its boundary. Think of it like a spoke on a bicycle wheel extending out to the tire.

• The diameter, on the other hand, is the longest distance across the circle, stretching from one end to the other, passing through the center.

The relationship between these two is straightforward: the diameter is simply twice as long as the radius. Knowing one of these values gives you the ability to calculate the other. This conversion factor is invaluable when solving problems related to circles, especially when determining the circumference. So, whenever you come across a circle-related task, remember the central role these measurements play.

Performing Mathematical Calculations for Circumference

Mathematical calculations are essential when finding the circumference of a circle, especially in practical scenarios like the bicycle wheel problem you've encountered.The formula to find the circumference is straightforward: \( C = 2 \pi r \), where \( r \) is the radius. This formula is derived from the fact that the circumference is just a little over three times the diameter, due to the constant \( \pi \). In our example, since the radius is given as 26.75 inches, calculating the circumference involves substituting the given value into our formula:\[ C = 2 \pi (26.75) \]To get an approximate answer, you can substitute \( \pi \) with 3.1416. This yields a calculated circumference of about 168.1193 inches, showing how far the wheel travels in one revolution. Rounding this number gives you a neat whole number, which in practical situations, simplifies the understanding of distance covered. Hence, by mastering these calculations, not only are you solving a problem, but you're also empowering yourself with a deeper understanding of mathematical relationships.