if a bicycle has 26 inch diameter wheels, the front chain drive has a radius of 2.2 inches, and the back drive has a radius of 3 inches. how far does the bicycle travel every one rotation of the cranks (pedals)?
The circumference is $\displaystyle \pi$ times diameter. So one rotation of the pedals will cause the front chain drive to complete one full turn, so it will move the chain the same distance as it's circumference. So it will move the chain $\displaystyle 2.2\pi$ inches. Then the back drive has a circumference of $\displaystyle 3\pi$ inches, but it only turns $\displaystyle 2.2\pi$ inches. So it turns $\displaystyle \frac{2.2\pi}{3\pi} = \frac{11}{15}$ of a full rotation. Since the wheel turns at the same rate as the back chain drive, the wheel will turn $\displaystyle \frac{11}{15}$ of a full turn as well. And since the back wheel is 26 inches in diameter, it's circumference is $\displaystyle 26\pi$ inches. If it turned a full rotation, it would go this distance along the ground, but it is only giong $\displaystyle \frac{11}{15}$ of this distance, so it is going $\displaystyle \frac{11}{15}*26\pi = \frac{286}{15}\pi$ inches. Or approximately 59.8997 inches.
...unless I misunderstood.
I don't want to pick at you but there is a slight difference between your description and your calculations. If the radius is 2.2 '' then the circumference is $\displaystyle 2 \cdot 2.2\pi$''.
When calculating the proportions the factor 2 cancels out so that your final result is correct.
Good call, I didn't catch the switch from diameter on the wheel to radius on the gear.
so then:
$\displaystyle C=2r\pi$
Front gear:
$\displaystyle C=2(2.2)\pi = 4.4\pi$
Rear gear:
$\displaystyle C=2(3)\pi = 6\pi$
Ratio of turn on rear gear
$\displaystyle C=\frac{4.4\pi}{6\pi} = \frac{11}{15}$
Rear wheel turns the same portion of a full turn as the rear gear
$\displaystyle \frac{11}{15}$
A full turn would be it's circumference
$\displaystyle C=2(3)\pi = 26\pi$ (because it is given in diameter rather than radius as the gears are)
Ratio of a full turn
$\displaystyle \frac{11}{15}*26\pi = \frac{286\pi}{15}$ inches
Well, I ended up with the same answer because the 2's ended up canceling out in the ratio. Is this what you came up with earboth? I don't have much confidence in my answer.
The answer given, 58.8997 should be 59.8997
Also, the response from earboth sez:
"When calculating the proportions the factor 2 cancels out so that your final result is correct."
The word "proportions" here should be the word "ratio".
Picky, indeed.
Greetings,
Montedoro