# Thread: Combinations of Functions

1. ## Combinations of Functions

Hi there. I'm having a bit of trouble with a word problem.
The question asks that you develop a formula predicting the velocity in km/h for a bicyclist
which is dependent on three independent factors
- the speed at which the cyclist pedals to turn the front gear (in rotations per minute)
- the gear ratio from the front gear to the rear wheel (ratio between #teeth on the front gear compared to #teeth on the rear gear)
- size of the rear wheel (diameter of wheel)

I thought that the formula would be something along the lines of
V=f*(a/b)*d
where f=rpm of front gear
a/b = ratio of teeth on front gear to back gear
d = diameter of wheel

But part b asks:
if a cyclist pedals at 50rpm with 42-toothed gear on the front, a 14-toothed gear on the back, and a tire with a diameter of 26 inches, how fast will the cyclist travel in km/h?

When i subbed these values into my equation I got an answer that I figure couldn't be right (0.00165km/h)

Can anyone shed some light for me?
Thanks a lot

2. Originally Posted by cazury
Hi there. I'm having a bit of trouble with a word problem.
The question asks that you develop a formula predicting the velocity in km/h for a bicyclist
which is dependent on three independent factors
- the speed at which the cyclist pedals to turn the front gear (in rotations per minute)
- the gear ratio from the front gear to the rear wheel (ratio between #teeth on the front gear compared to #teeth on the rear gear)
- size of the rear wheel (diameter of wheel)

I thought that the formula would be something along the lines of
V=f*(a/b)*d*pi
where f=rpm of front gear
a/b = ratio of teeth on front gear to back gear
d = diameter of wheel

But part b asks:
if a cyclist pedals at 50rpm with 42-toothed gear on the front, a 14-toothed gear on the back, and a tire with a diameter of 26 inches, how fast will the cyclist travel in km/h?

When i subbed these values into my equation I got an answer that I figure couldn't be right (0.00165km/h)

Can anyone shed some light for me?
Thanks a lot
If I use your values I'll get:

$v = 50 \cdot \dfrac{42}{14} \cdot \pi \cdot \underbrace{26 \cdot 0.0254}_{in \to m} = 311.2\ \frac m{min}$

In 1 hour the distance is:

$311.2\ \frac m{min} \cdot 60 \ min= 18.67 \ km$

Therefore the average speed of the bicyclist is 18.67 km/h.

3. Ohh that makes complete sense.
Thank you so much