1. ## hard word problem

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 teeth rear wheel (a ratio between the number of teeth on the front gear compared to 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 50rpm 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?

2. Originally Posted by math123456
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:
...
b) How fast will a cyclist travel (in km/h) who is pedalling at 50rpm with a 42-toothed gear on the front, a 14-toothed gear on the back, and a tire with a diameter of 26 inches?
...
I'll do #b). You can answer the remaining questions if you use my calculations:

1. $\displaystyle 50\ rpm = 3000\ rph$

2. $\displaystyle ratio = \dfrac{front-gear}{rear-gear}=\dfrac{42}{14}=3$

3. circumference of the rear wheel: $\displaystyle c = \dfrac{2\cdot \pi \cdot 13\ in \cdot 2.54\ \frac{cm}{in}}{100,000\ \frac{cm}{km}} = 0.0020747\ \frac{km}{1\ rev}$

4. speed $\displaystyle v = 3000\ rph \cdot \underbrace{3}_{ratio} \cdot 0.0020747\ \frac{km}{1\ rev} = 18.67\ \frac{km}{h}$