a 21-speed bike has varying top speeds dependent on the gear it's in. the bike's tire diameter is 26 inches. the bike can pedal at a maximum of 130 revolutions per minute. the front chain wheel has sprokets of 24, 32, and 40 teeth. the rear cassette sprokets have 28, 24, 21, 18, 16, 14, 12 teeth. the bike's gears are labeled 1-21. gears 1-7 associated with the smallest front chain wheel's sproket. clearly communicate the top speed (in miles per hour) in each gear.

The fastest speed of the bike at a fixed pedal speed is achieved at the highest front/back sprocket or "gear" ratio of 40/12. Multiply that by 120 rev/min and pi*Diameter to get the top speed

(130 rev/min)*(40/12)*pi*(26/12 feet) = 2949.6 ft/min = 49.2 ft/s = 33.5 miles/h

Speeds with other gear combinations can be calculated similarly. The lowest speed is
(130 rev/min)*(24/28)*pi*(26/12 ft) = 758.5 ft/min = 8.6 mph at the same pedal speed. The actual speed will be slower still if the rider reduces the pedal rpm, as most riders do in the lowest gear

To calculate the top speed in each gear of a 21-speed bike, you need to know the gear ratio, which is determined by the combination of front chainwheel and rear cassette sprockets being used. The gear ratio represents the number of times the rear wheel turns for every rotation of the pedals. Given the gear setup provided, here's how you can find the top speed in miles per hour for each gear:

1. Determine the gear ratio for each gear:

- Gear 1: The smallest front chainwheel with the largest rear cassette sprocket. The gear ratio is calculated by dividing the number of teeth on the front chainwheel (24) by the number of teeth on the rear cassette sprocket (28): 24 / 28.
- Gear 2: Same front chainwheel, different rear cassette sprocket. Calculate the gear ratio using the front chainwheel (24) and the next smallest rear cassette sprocket (24): 24 / 24.
- Gear 3: Same front chainwheel, different rear cassette sprocket. Calculate the gear ratio using the front chainwheel (24) and the next smallest rear cassette sprocket (21): 24 / 21.
- Continue this calculation for all remaining gears, using the appropriate front chainwheel and rear cassette sprocket combinations.

2. Calculate the distance traveled per pedal revolution:

- To find out how far the bike travels for each pedal revolution, you need to know the circumference of the bike's tire. For a 26-inch tire, the circumference can be calculated using the formula: 2 * π * radius, where the radius is the tire diameter divided by 2.
- Convert the tire circumference to miles by dividing it by 12 (to convert inches to feet) and then by 5,280 (to convert feet to miles).

3. Calculate the top speed in miles per hour:

- Multiply the gear ratio by the distance traveled per pedal revolution. This will give you the speed in miles per pedal revolution.
- Multiply the speed in miles per pedal revolution by the maximum pedal revolutions per minute (130) to get the top speed in miles per hour.

By following these steps for each gear on the bike, you can determine the top speed (in miles per hour) in each gear.