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 travels \( 600\pi \) feet.

Step 1: Understanding the Problem

To find out how far the wheel rolls, we need to calculate the distance it covers with each revolution and then multiply this by the total number of revolutions. This involves using the circumference of the wheel.

Step 2: Calculate the Circumference of the Wheel

The circumference of a circle is given by the formula \( C = 2\pi r \), where \( r \) is the radius. Here, \( r = 2 \) feet. So, the circumference \( C = 2\pi \times 2 = 4\pi \) feet.

Step 3: Calculate the Total Distance for 150 Revolutions

The total distance the wheel travels is equal to the number of revolutions multiplied by the circumference of the wheel. Thus, the total distance \( = 150 \times 4\pi \) feet.

Step 4: Final Calculation

Calculate \( 150 \times 4\pi \) to find the total distance: \( 600\pi \) feet. This is the final distance traveled by the wheel after 150 revolutions.

Key Concepts

Revolutions

When we talk about revolutions in the context of wheels or circles, we are referring to complete turns around a central point. Each revolution signifies one complete rotation. For a wheel, one revolution means it has rolled all the way around its circumference.
This concept is important because it helps in calculating the distance a wheel can travel if we know how many times it rotates fully.
In practical scenarios, such as this exercise, understanding the number of revolutions helps us determine how far an object can move.

• One revolution clockwise or counterclockwise is the same in terms of distance covered.
• It's a repeated action that involves rolling or rotating back to the original point.
• Projectile mechanics or crank mechanisms are often examples related to revolutions.

Circle

A circle is a simple shape but essential in understanding how objects move and roll. It is defined as a set of points in a plane that are equidistant from a given point called the center. The distance from the center to any point on the circle is the radius.
In mathematical terms, the circumference of a circle, which is the total distance around it, can help us understand how far a wheel travels in one complete revolution.

• The radius is half the diameter, which is the maximum distance across the circle.
• The circle is symmetrical; all points on its edge have the same distance to the center.
• It doesn't have any edges or corners, making it perfect for objects that roll.

Understanding these properties is key to solving problems involving motion using wheels or circular paths.
Knowing about circles is fundamental if you're diving into geometry or any wheel-based motion analysis.

Distance Calculation

Calculating distance is crucial in numerous fields, from engineering to everyday scenarios. In this context, knowing the circumference (distance around the wheel) is essential because it allows you to calculate how far a circular object like a wheel can travel.
Here's a step-by-step way to break down distance calculation:

• First, know the formula for circumference, which is \( C = 2\pi r \). For our problem, using a radius of 2 feet gives us \( C = 4\pi \) feet.
• Recognize that each full revolution covers a distance equal to the circumference.
• Multiply the number of revolutions by the circumference to find total distance. With 150 revolutions, multiply \( 150 \times 4\pi \), giving \( 600\pi \) feet travelled.

The calculated product \( 600\pi \) tells you the precise total distance covered in simple and clear terms.
Mastering distance calculation makes understanding more complex problems easier, and it's a foundational principle in both mathematics and physics.