Problem 36 - Calculus Early Transcendentals: Pearson New International Edition

Question

. How far does a wheel of radius 2 feet roll along level ground in making 150 revolutions?

Step-by-Step Solution

Answer

The wheel rolls 600\pi feet.

Step 1: Determine the Circumference of the Wheel

To find the distance the wheel rolls in one complete revolution, we first need to calculate the circumference of the wheel. The formula for the circumference of a circle is \( C = 2\pi r \), where \( r \) is the radius of the circle. Here, the radius \( r \) is 2 feet. So the circumference \( C = 2\pi \times 2 = 4\pi \) feet.

Step 2: Calculate the Total Distance for 150 Revolutions

Now that we have the circumference of the wheel, we can determine how far the wheel rolls after 150 revolutions. The total distance rolled is the product of the number of revolutions and the circumference of the wheel: \( \text{Total Distance} = 150 \times 4\pi \).

Step 3: Simplify the Expression for Total Distance

Multiply the number of revolutions by the circumference of the wheel: \( 150 \times 4\pi = 600\pi \). Therefore, the wheel rolls a total distance of \( 600\pi \) feet.

Key Concepts

Circumference of a Circle

The circumference of a circle is the distance around it, much like the perimeter for polygons. Imagine a tightly wrapped piece of string around a circular object. Unwrap it, lay it out, and its length is the circumference. To calculate the circumference, we use the formula:

• \( C = 2\pi r \)

Where:

• \( C \) is the circumference
• \( \pi \) is approximately 3.14159, a special mathematical constant
• \( r \) is the radius of the circle

For our wheel with a radius of 2 feet, the formula becomes \( C = 2\pi \times 2 \), resulting in a circumference of \( 4\pi \) feet. This simple yet powerful formula allows us to understand how far the wheel travels with each revolution.

Circle Radius

A radius is a line from the center of a circle to any point on its edge. It's half the diameter and gives us vital information about the circle's size.

• The radius is central to calculating both the circumference and the area of a circle, but in this scenario, we use it for the circumference.
• Knowing the radius, we can find out how large the circle is, which directly contributes to the distance the wheel will travel in one revolution.

For example, with a radius of 2 feet for our wheel, we can determine the full distance covered by the wheel each time it completes one circle around its axis using the circumference formula. The radius is crucial for these types of calculations, as it directly affects the results.

Mathematical Calculation

Mathematical calculations are steps or processes used to derive specific values or solve problems like finding the distance a wheel covers. Here's how we solve our problem:

• First, determine the circumference using the circle's radius.
• Next, multiply the circumference by the number of revolutions to get the total distance.

For our wheel:

• Circumference \( = 4\pi \) feet.
• Multiply by 150 revolutions: \( 150 \times 4\pi = 600\pi \) feet.

The result, \( 600\pi \), gives us the complete distance covered by the wheel on level ground. This methodical approach ensures accuracy. Mathematical calculations require a clear understanding of how to use formulas and operations like multiplication to reach the correct conclusions.