Problem 70 - Precalculus Mathematics for Calculus

Question

How many revolutions will a car wheel of diameter 30 in. make as the car travels a distance of one mile?

Step-by-Step Solution

Answer

The wheel makes approximately 672 revolutions.

Step 1: Calculate the Circumference of the Wheel

The circumference of a circle is calculated using the formula \( C = \pi \times d \), where \( d \) is the diameter. For a wheel with a diameter of 30 inches, the circumference \( C \) is \( C = \pi \times 30 \approx 94.25 \, \text{inches} \).

Step 2: Convert Miles to Inches

There are 63,360 inches in a mile (since 1 mile = 5,280 feet and 1 foot = 12 inches). Therefore, 1 mile is equivalent to 63,360 inches.

Step 3: Calculate the Number of Revolutions

To find the number of revolutions, divide the total distance traveled by the circumference of the wheel. This can be calculated by \( \frac{63,360}{94.25} \approx 672.04 \). This means the wheel will make approximately 672 revolutions.

Key Concepts

Revolutions

When we talk about revolutions of a wheel, we refer to the number of times the wheel completely turns around its axis. This concept applies to any circular object moving along a path, like a car wheel. Each complete "turn" or revolution means the point that started at the ground makes a full circle and returns to the ground.
To find out how many revolutions a wheel makes while traveling a certain distance, it's crucial to know the wheel's circumference. The circumference is the distance around the wheel, so the number of revolutions is the total distance traveled divided by the circumference.

• If you know the total distance and the circumference, you can easily find the revolutions.
• The formula is: \[ \text{Revolutions} = \frac{\text{Total Distance Traveled}}{\text{Circumference of the Wheel}} \]

Understanding revolutions helps with tasks like determining how far a vehicle travels given the number of turns the wheels make. This concept is fundamental in the maintenance and monitoring of vehicles.

Unit Conversion

Unit conversion is an essential process in solving many mathematical problems. It involves changing a measurement from one unit to another, facilitating easier arithmetic or comparison.
In our exercise, we need to convert miles to inches to match the unit of the wheel's diameter, which is given in inches.
To convert miles to inches:

• First, know the basic conversions: 1 mile = 5,280 feet and 1 foot = 12 inches.
• Thus, 1 mile = 5,280 feet × 12 inches/foot = 63,360 inches.

These conversions are helpful because they maintain consistency across the different measures we are using, allowing for accurate calculations. It’s a valuable skill in math and helps streamline solving real-world problems by transforming all quantities to compatible units before performing operations.

Distance Calculation

Distance calculation, particularly in the context of wheel revolutions, involves computing how far an object travels given certain parameters.
With wheels, the most straightforward calculation to determine distance involves multiplying the number of revolutions by the circumference of the wheel.
The formula looks like this:

• Distance traveled = Revolutions × Circumference.

In practical terms, this calculation can help verify distance accuracy if you have a known number of wheel rotations. Conversely, if you already know the travel distance and the wheel's circumference, you can find out how many times the wheel turned (the number of revolutions).
This is critical not only in vehicle monitoring but also in a variety of mechanical and engineering applications. Understanding how to calculate distances using these principles makes it easier to predict movements and design systems effectively.