**How fast is the bicycle traveling if the rear wheel is rotating at a rate of 260 revolutions per minute?**

To determine how fast the bicycle is traveling given that the rear wheel is rotating at 260 revolutions per minute (rpm), we can follow these steps:

### Step 1: Calculate the Circumference of the Wheel
The circumference \( C \) of the wheel, which is the distance the bicycle travels in one full rotation of the wheel, is calculated as:
\[
C = \pi \times d
\]
Given that the diameter \( d \) of the wheel is 26 inches:
\[
C = \pi \times 26 \approx 81.68 \text{ inches}
\]

### Step 2: Calculate the Distance Traveled per Minute
The distance traveled per minute is the product of the number of revolutions per minute (rpm) and the circumference of the wheel:
\[
\text{Distance per minute} = \text{rpm} \times C
\]
Substituting the given rpm:
\[
\text{Distance per minute} = 260 \times 81.68 \approx 21,237.6 \text{ inches per minute}
\]

### Step 3: Convert Distance per Minute to a More Common Unit (Miles per Hour)
First, convert inches per minute to miles per hour (mph).

- There are 12 inches in a foot, and 5280 feet in a mile, so there are:
\[
5280 \times 12 = 63,360 \text{ inches in a mile}
\]
- To convert inches per minute to miles per hour, we use:
\[
\text{Speed (mph)} = \frac{\text{Distance per minute (in inches)} \times 60 \text{ minutes}}{63,360 \text{ inches per mile}}
\]
\[
\text{Speed (mph)} = \frac{21,237.6 \times 60}{63,360} \approx 20.1 \text{ mph}
\]

### Conclusion:
The bicycle is traveling at approximately **20.1 miles per hour** when the rear wheel is rotating at 260 revolutions per minute.

Would you like more details or have any questions about this calculation?

Here are 5 related questions for further exploration:

1. What would the speed be if the wheel diameter were increased to 28 inches?
2. How fast is the bicycle traveling if the rear wheel rotates at 300 rpm?
3. What is the rpm if the bicycle is traveling at 25 mph?
4. How does changing the gear ratio affect the speed of the bicycle at a given rpm?
5. How far will the bicycle travel

Learn how to calculate the speed of a bicycle given the rear wheel RPM. This problem involves calculating the circumference of the wheel, converting units from inches per minute to miles per hour, and applying basic geometric principles. Suitable for Grades 7-9.

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