Problem 69 - Precalculus Mathematics for Calculus

Question

A car's wheels are 28 in, in diameter. How far (in miles) will the car travel if its wheels revolve \(10,000\) times without slipping?

Step-by-Step Solution

Answer

The car will travel approximately 1.387 miles.

Step 1: Find the Circumference of the Wheel

The circumference of the wheel is the distance it travels in one complete revolution. Use the formula for the circumference of a circle, which is \( C = \pi \times d \), where \( d \) is the diameter of the wheel. Here, \( d = 28 \) inches. Hence, the circumference is \( C = \pi \times 28 = 28\pi \) inches.

Step 2: Calculate Total Distance in Inches

To find the total distance the car travels after 10,000 revolutions, multiply the circumference by the number of revolutions. The total distance is \( 28000\pi \) inches.

Step 3: Convert Inches to Miles

Convert the total distance from inches to miles. There are 12 inches in a foot and 5280 feet in a mile. Therefore, there are \( 12 \times 5280 = 63360 \) inches in a mile. The total distance in miles is \( \frac{28000\pi}{63360} \approx 1.387 \) miles.

Key Concepts

Unit Conversion

When working with measurements, unit conversion is a key skill. It allows you to switch from one unit to another. Understanding the relationships between units is essential.

• For our exercise, we start with a measurement in inches and convert it to miles.
• We use the fact that there are 12 inches in a foot and 5280 feet in a mile.
• Multiplying these values gives us 63360 inches per mile. This conversion factor is crucial.

To convert from inches to miles, divide the number of inches by the conversion factor. In this case, the equation is:\[\text{Miles} = \frac{\text{Inches}}{63360}\]Converting units is about keeping track of the relationship between different units and calculating accurately to maintain the correct scale.

Distance Traveled

The concept of distance traveled in this problem revolves around understanding the wheel's movement. Here's how:

• The distance traveled in one wheel revolution is the same as its circumference.
• Multiply the circumference by the total number of revolutions to get the total distance traveled.

In the problem, the wheel's circumference is \(28\pi\) inches. Since the wheels revolve 10,000 times, we calculate:\[\text{Total Distance} = 28000\pi \text{ inches}\]This insightful calculation shows how multiple small revolutions add up to a larger total distance.

Mathematical Problem Solving

Mathematical problem-solving involves breaking down complex issues into manageable steps. Using logical reasoning and mathematical principles is the foundation. Here's how we approached this exercise:

• Start by identifying what you know: the wheel's diameter and the number of revolutions.
• Calculate the circumference, which helps understand one full revolution.
• Derive the total distance in inches by multiplying the circumference by the number of revolutions.
• Convert the distance from inches to miles to provide a familiar measurement context.

Each step connects methodically to form a complete solution. This approach can be applied to various mathematical challenges, emphasizing clarity and logical progression.