trigonometry

The sprockets and chain of a bicycle are shown in the figure. The pedal sprocket has a radius of 4 in., the wheel sprocket a radius of 2 in., and the wheel a radius of 13 in. The cyclist pedals at 45 rpm.

(a) Find the angular speed of the wheel sprocket.
rad/min

(b) Find the speed of the bicycle. (Assume that the wheel turns at the same rate as the wheel sprocket. Round your answer to one decimal place.)
mi/h

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  1. angular velocity:

    1 rotation = 2π radians
    45 rotations = 90π radians

    angular velocity = 90π radians/minute

    Since circumference is linear and we are told the radius of the pedal sprocket is twice that of the wheel sprocket, for every turn of the pedal, the wheel will make 2 rotations
    So in 1 minute the wheel turns 90 times
    circumference of wheel = 2π(13) = 26π inches
    distance covered in 1 minute = 90(26π) = 2340π inches

    speed = 2340π inches/min
    there are 12(5280) inches in 1 mile
    and 60 minutes in 1 hour

    speed = 2340π(60/(12(5280)) miles/hr
    = appr 7.0 mph

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