A bicycle with 18in 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 180 rpm.

Let's break it down into simple steps.

The linear speed of the bicycle is the distance traveled by the bike in a given amount of time. In other words, speed is calculated by distance divided by time. In this case, we want to find the speed in inches per minute. To calculate this, we need to determine how far the bike travels in one revolution of the pedals and then multiply by total revolutions per minute.

The circumference of a circle (bike wheels) is calculated using the formula: circumference = 2 * π * radius.

For the wheel the radius is given as 18/2=9 inches. So the circumference of the wheel is 2 * π * 9 = 18 * π inches.

Now, when the cyclist pedals once (one complete revolution), the chain moves the same distance as the circumference of the wheel. So, to find the linear speed in inches per minute, we need to multiply the distance traveled in one revolution by the number of revolutions per minute (rpm). So the formula becomes:

Linear speed (inches/minute) = Circumference of the wheel (in inches) * Revolutions per minute

To convert the linear speed from inches per minute to miles per hour, we need to make two conversions:

- Convert inches to miles
- Convert minutes to hours

There are 12 inches in a foot, and 5,280 feet in a mile. So to convert inches to miles, we divide the number of inches by (12 * 5280).

Similarly, there are 60 minutes in an hour. So to convert minutes to hours, we divide the number of minutes by 60.

So, to find the speed in miles per hour, we divide the linear speed (in inches per minute) by (12 * 5280) to convert it to miles, and then multiply by 60 to convert minutes to hours.

Speed (miles per hour) = (Linear speed in inches per minute / (12 * 5280)) * 60

That's it!

For every cycle of the 6 inch sprocket the 4 inch goes around 6/4 = 1.5 times. In other words, every cycle of the front sprocket equals 1.5 cycles of the back sprocket.

Therefore:

Linear speed of the bicycle = 180 x 1.5 x π X 18 = 15,268 inches/ min (linear speed of the bicycle)

Now,

(15268/12)*60/5280) = 14.45 mph

So, bike's speed is 14.45 mph.