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Question 1107119: A bicycle has two sets of gears, one at the pedals and one at the rear wheel. A typical 10 speed bike has two gears on the chain wheel and five on the rear wheel. The speed at which a bicycle travels depends on three independent factors: The speed which the cyclist pedals to turn the front gear measured in rotations per minute; the gear ratio from the front gear to the rear wheel; size of the rear wheel measured as the diameter of the wheel.
a) Develop a formula that predicts the speed 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 the cyclist pedal more slowly but maintain the same speed?
This question is in a chapter called Combinations of Functions from a Pre-Calculus textbook. From some of the examples given in the textbook I believe the correct answer involves an sinusoidal/trig equation in the form
and 
Final result 
I'm trying to picture the equation on a graph. Not sure, but would the amplitude have something to do with the rear wheel's diameter of 26 inches and the period with the 50 RPM?
Thank you in advance
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! not sure if this is right, but it's worth a shot.
the diameter of the rear wheel is equal to 26 inches.
the circumference of the rear wheel is therefore pi * 26 = 81.68140899
the rear wheel will travel 81.68140899 inches when the rear wheel does one complete rotation.
now the front gear has 48 teeth and the rear gear has 14 teeth.
that means the rear gear rotates 3 times for every 1 time of the front gear.
if the bicyclist is rotating the front gear 50 times a minute, then the rear gear must be rotating 150 times a minute.
since the rear gear and the rear wheel rotate at the same rate, this means that the rear wheel is rotating 150 times a minute.
this means that the bike is moving forward 150 * 81.68140899 = 12,252.21235 inches per minute.
divide that by 12 and you get 1,021.017612 feet per minute.
multiply that by 60 and you get 61,261.05675 feet per hour.
divide that by 3280.84 and you get 18.6723695 kilometers per hour.
the formula can't be graphed very easily because there are a lot of factors that determine how fast the bicycle will go.
the formula i developed is as follows:
S = F/B * pi * D * RPM * 60 / (12 * 3280.84)
S is the speed that the bike is traveling in kilometers per hour.
F is the number of teeth in the front gear.
B is the number of teeth in the back gear.
pi is equal to 3.142592654
D is the diameter of the rear wheel.
RPM is the number of revolutions per minute of the front gear.
60 is the number of minutes in an hour.
12 is the number of inches in a foot.
3280.84 is the number of feet in a kilometer.
in your problem, the front gear has 42 teeth and the rear gear has 14 teeth and the cyclist is pedaling at 50 RPM and the diameter of the rear wheel is 26 inches.
the formula becomes S = 42/14 * pi * 26 * 50 * 60 / (12 * 3280.84)
solve for S to get S = 18.6723695 kilometer per hour.
if the bicyclist wants to go faster while pedaling at the same RPM, then he can do any of the following.
1.
get a front gear with more teeth.
2.
get a back gear with less teeth.
3.
get a rear wheel with a larger diameter.
your equation can be graphed, but you have to make most of the variables constants and only make of the factors the variable of interest.
the following graphs will show you what i mean.
the first graph assume all factors are constant except the rpm.
therefore, you let y = S, which represents the speed of the bike in kilometers per second and you let x equal the number of revolutions per minute of the front gear.
the formula will be y = S = 42/14 * pi * 26 * x * 60 / (12 * 3280.84)
the graph will look like this:
this graph will show you the speed of the bicycle for different revolutions per minute. x is the rpm.
if you want to keep the revolutions per minute constant but vary the front teeth to back teeth ratio, then the formula will be y = x * pi * 26 * 50 * 60 / (12 * 3280.84)
the graph will look like this:
this graph will show you the speed of the bicycle with different front gear to back gear ratios.
x is the front gear to back gear ratio.
you can do the same with the other constants by making them the variable of interest.
if you use the formula where x is the rpm of the bike, and you just look at what happens at 50 rpm, then you can play with the other constants to see what happens when you change each of the factors.
for example:
make the ratio of the front gear to back gear 84/14 and the speed will double at 50 RPM.
make the ratio of the front gear to back gear 42/28 and the speed will be cut in half.
make the diameter of the rear wheel double and the speed will double.
the overall equation, if you're not concerned about graphing it on a two dimensional plane, will be:
S = F/B * pi * D * RPM * 60 / (12 * 3280.84)
since pi * 60 / (12 * 3280.84) will always be the same, you can combine those into one factor.
your formula will become:
S = F/B * D * RPM * .0047877871
when F = 42 and B = 14 and D = 26 and RPM = 50, you get S = 18.6723695, same as before, as you should, since all you did was combine all the constants into one factor.
this may or may not be what you are looking for, but it does appear to provide the answer for translating all the different factors into the speed of the bike in kilometers per hour.
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