Combining functions word problem?  TOPIC_SOLVED

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Combining functions word problem?

Postby Math Novice on Fri Nov 12, 2010 6:37 pm

A bicycle has two sets of gears, one at the pedals and one at the rear wheel. A typical 10-speed bicycle has two gears on the chain wheel and five gears on the rear wheel. The speed at which a bicycle travels depends on three independent factors:
- The first is the speed at which the cyclist pedals to turn the front gear (measured in rotations per minute).
- The second is the gear ratio from the front gear to the rear wheel (a ratio between the number of teeth on the rear gear).
- The third is the size of the rear wheel (measured as the diameter of the wheel).
a)Develop a formula that predicts the velocity in km/h for a bicyclist.
b)How fast will a cyclist travel (in km/h) who is pedalling at 50 rpm with a 42-toothed gear on the front, a 14-toothed gear on the back, and a tire with a diameter of 26 inches?
c)In what ways can a cyclist pedal more slowly but maintain the same speed?

I appreciate all the help I can get, I have no idea how to start solving this problem. Also, there doesn't seem to be anything about combining functions on purple math, unless I might have missed it.
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  TOPIC_SOLVED

Postby stapel_eliz on Sat Nov 13, 2010 12:10 am

Math Novice wrote:a)Develop a formula that predicts the velocity in km/h for a bicyclist.

If the cyclist pedals at a rate of "x" rotation per minute, then how many times does the front gear rotate per minute?

Given a gearing ratio of "a:b" (that is, given that one rotation of the front gear is a/b rotations of the rear gear), how many times does the rear tire rotate in one minute?

Given a diameter of "d" inches, what is the circumference of the rear tire, in inches?

Given that the rear tire has the given number of rotations per minute, each covering the circumference's number of inches, how many (linear) inches does the rear tire cover in one minute?

Multiply this expression by a conversion factor to go from "inches per minute" to "kilometers per hour". Set the result equal to "S" to create your "speed" equation.

Math Novice wrote:b)How fast will a cyclist travel (in km/h) who is pedalling at 50 rpm with a 42-toothed gear on the front, a 14-toothed gear on the back, and a tire with a diameter of 26 inches?

Plug the given values into your equation, and simplify to find the speed.

Math Novice wrote:c)In what ways can a cyclist pedal more slowly but maintain the same speed?

Plug the speed you just found, along with the given diameter, into your equation. Fiddle with the pedal speeds and gearing ratios to see what you can come up with. (Did they give you any rules for the gears, to allow you to limit your choices?)

Math Novice wrote:...there doesn't seem to be anything about combining functions on purple math, unless I might have missed it.

Purplemath has lessons on "operations on functions and function composition. What do you mean by "combining" functions? :confused:
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