Problem 69 - Algebra and Trigonometry

Question

Travel Distance 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 1.39 miles.

Step 1: Find the circumference of the wheel

The circumference of a circle (wheel) can be calculated with the formula \( C = \pi \times d \), where \( d \) is the diameter. Given the diameter is 28 inches, the circumference of each wheel is \( C = \pi \times 28 \) inches.

Step 2: Calculate the total distance in inches

To find the total distance traveled, multiply the circumference of the wheel by the number of revolutions: Total distance (in inches) \( = C \times 10000 \). Therefore, the total distance is \( 28000 \pi \) inches.

Step 3: Convert inches to miles

First, convert inches to feet by dividing by 12, then convert feet to miles by dividing by 5280 (since there are 5280 feet in a mile). So, the formula becomes: \[ \text{Distance in miles} = \frac{28000 \pi}{12 \times 5280} \approx 1.39 \text{ miles}. \]

Key Concepts

Distance Conversion

Distance conversion is a key skill in many mathematics and physics problems, especially when different units are used in the same problem. In this exercise, we started with a distance measured in inches and needed to convert it to miles. Understanding how units relate to each other is vital.

• There are 12 inches in a foot.
• There are 5280 feet in a mile, which is a very common conversion factor used in problems involving distance.

To convert between these units, set up a series of multiplication or division steps. First, convert the total distance in inches to feet by dividing the inches by 12, then convert the resulting feet to miles by dividing by 5280. This method ensures accurate and complete conversion across units.

Wheel Revolutions

Wheel revolutions are a practical concept used to determine how far a vehicle travels. Each complete turn of the wheel (a revolution) covers a distance equal to the wheel's circumference. In this exercise, we calculated how far a vehicle moves based on its wheels' revolutions.

• The circumference is how far one revolution moves the wheel.
• Multiply the circumference by the number of revolutions to find the total distance traveled.

The given problem involved 10,000 revolutions, each contributing to the total distance. By understanding wheel revolutions, we can better relate physical wheel movement to actual distances traveled in scenarios like driving or tracking wheel-based devices.

Diameter and Radius

The diameter and radius are fundamental elements of a circle, influencing calculations such as area and circumference. These values help identify the size of a circle component like a wheel.

• The diameter is the distance across the circle through its center.
• The radius is half the diameter, running from the center to the edge of the circle.

For this exercise, the wheel's diameter was given as 28 inches. Using this diameter, the circumference can be found with the formula: \[ C = \pi \times d \] where \( d \) is the diameter of the circle. This formula helped us find the distance covered in one wheel revolution.

Unit Measurement Conversion

Unit measurement conversion is an essential math skill that allows us to navigate problems involving different measurement systems or units. Whether dealing with imperial (foot, inch, mile) or metric systems, the ability to convert units accurately is critical.

• Understand the relationships: 1 foot = 12 inches, and 1 mile = 5280 feet.
• Use division or multiplication to convert units: from smaller to larger units, divide; from larger to smaller units, multiply.

In our solution, converting from inches to miles was done in two steps: first converting inches into feet and then feet into miles. Practicing these conversions aids in understanding larger distance calculations and ensures precision in practical applications such as measuring travel distances.