Bicycle angular and Linear speed problem

Question

A bicycle with 20-in.-diameter wheels has its gears set so that the chain has a 6-in. radius on the front sprocket and 4-in. radius on the rear sprocket. The cyclist pedals at 175 rpm.

Find the linear speed of the bicycle in in/min (correct to at least two decimal places)

How fast is the bike moving in mph (to two decimal places)?

Answer 1

The front gear has a radius of 6 inches. Each revolution drives it once around its circumference, which is 6 × 2 × π = 12π. It is revolving at 175 rpms, so it is moving 175 × 12π = 2,100π inches per minute.

Divide this by 4 × 2 × π, the circumference of the rear gear. 2100π/(8π) = 262.5 revolutions per minute.

This means the rear wheel is moving at 262.5 rpms as well. The rear wheel has a diameter of 20 inches. Its circumference is 20π inches.

At 262.5 rpms, the bike will travel 262.5 × 20π inches per minute, or

16,493.36 inches per minute.

This is about 15.62 miles per hour.