Problem 72 - Precalculus

Question

A bicycle has wheels 28 inches in diameter. A tachometer determines that the wheels are rotating at 180 \(\mathrm{RPM}\) (revolutions per minute). Find the speed the bicycle is travelling down the road.

Step-by-Step Solution

Answer

The bicycle travels at approximately 15.01 mph.

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. Given that the diameter of the bicycle wheel is 28 inches, the circumference is \(C = \pi \times 28\).

Step 2: Express the Circumference in Inches

Calculate the actual number for the circumference by multiplying: \(C = \pi \times 28 \approx 87.96\) inches.

Step 3: Calculate the Distance Per Minute Traveled

Every revolution of the wheel moves the bicycle forward by the length of the circumference. Since the wheel rotates at 180 RPM, in one minute the bicycle covers \(87.96 \times 180\) inches.

Step 4: Convert Inches Per Minute to Miles Per Hour

First, convert the distance traveled per minute into distance traveled per hour by multiplying by 60 (since there are 60 minutes in an hour): \(87.96 \times 180 \times 60\). This gives us the distance in inches per hour. Then, convert inches to miles using the conversion factor: 1 mile = 63,360 inches. Thus, the speed in miles per hour is \(\frac{87.96 \times 180 \times 60}{63,360}\).

Step 5: Calculate the Speed in Miles Per Hour

Perform the final calculation to find the speed: \(\frac{87.96 \times 180 \times 60}{63,360} \approx 15.01\) mph.

Key Concepts

Circumference of a Circle

When determining how far a bicycle wheel will travel in one complete revolution, it's essential to understand the concept of the circumference of a circle. The circumference is the total distance around the circle. To calculate it, you use the formula: \( C = \pi \times d \). Here, \( \pi \) (Pi) is approximately 3.14159, and \( d \) is the diameter of the circle. For example, if a bicycle wheel has a diameter of 28 inches, simply multiply \( \pi \) by 28 to find the circumference. This measurement tells us how far the wheel will roll in a single turn.

Revolutions per Minute (RPM)

Revolutions per Minute (RPM) measures how many complete turns something makes in one minute. For a bicycle wheel, a higher RPM means the wheel is rotating faster, propelling the bike forward at a greater speed. Knowing the RPM is crucial for calculating speed, as it directly indicates how many times the wheel’s circumference moves the bicycle forward in a minute. In our scenario, with the wheel rotating at 180 RPM, it completes 180 full rotations every minute. Hence, in one minute, the wheel travels a distance equivalent to 180 times its circumference.

Unit Conversion

Often in physics and engineering, you have to convert between units to make calculations easier or more meaningful. In this case, the fundamental task is to change the speed from inches per minute to miles per hour. First, you need to find out how many inches the wheel travels in an hour by multiplying the distance per minute by 60. Yet, since miles per hour are more commonly used for speed, convert the total inches per hour to miles. Remember the key conversion factor: 1 mile equals 63,360 inches. Using this factor ensures your final answer is in the desired unit of measurement, making it useful for real-world applications.

Bicycle Wheel Diameter

The diameter of a bicycle wheel is the straight-line measurement passing through its center, from one side to the other. This value is critical because it influences both the circumference and the RPM-to-speed conversion. A larger diameter means a larger circumference, so the wheel covers more ground with each revolution. Conversely, smaller wheels have a smaller circumference and travel a shorter distance per revolution. Therefore, knowing the wheel diameter lets you accurately calculate speed when given the RPM. In our example, a 28-inch wheel diameter directly affects how we compute the total distance traveled and ultimately the speed of the bicycle.