Find the linear speed and how fast the bicycle is going.?

A bicycle with 20-in diameter wheels has its gears set so that the chain has a 7-in radius on the front sprocket and 4-in on the rear spocket. The cyclist pedals at 180 rpm.

So i tried multiplying the rpm by the diameter of the wheel but i don't know what to do with the sprockets and how they are related.

Please help.
 
convert revolutions into terms of pi...
(180 rev./1 min.) * (2pi rad/1 rev.) = (180 * 2pi)/1 min. = 360pi/1 min.

so the 7-in. sprocket is turning at 360pi per minute.

convert to linear velocity using v = rw, where w is angular velocity in radians per minute...

v = 7 in. * (360pi/min.) = 2520pi in./min.

the 7-in. sprocket is connected to the 4-in. sprocket, so use v = rw to convert to get the angular velocity of the 4-in. sprocket. First, solve for w.

v = rw
v/r = w

plug in the numbers...

(2520pi in./min)/4 in. = w = 630pi in./min.

the 4-in. sprocket is connected to the wheel, so use the formula v = rw to get the linear velocity of the wheel and thus the speed of the bike. The radius of the wheel is 10 in.

v = rw
v = 10 * 630pi = 6300pi in./min.

the bicycle is traveling at 6300pi in./min. To check if this makes sense, convert to mph...
(6300pi in./min.) * (60 min./1 hour) = 378000pi in./hour
(378000pi in./hour) * (1 ft./12 in.) = 31500pi ft./hour
(31000pi ft./hour) * (1 mile/5280 ft.) = 5.9659pi mile/hour
5.9659pi mile/hour = 18.74mph

18.74mph sounds reasonable.
 
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