# Approximation

### Approximation, conversions and back-of-the-napkin computations

• A Marathon Problem
```During the 2008 Flying Pig Marathon there was a fire on the route, resulting in a slight detour.  The
length of a marathon is usually 26.2 miles.  The detour added .2514 miles to the race distance.  Casey
Fein, a 24-year old runner, finished the race with a time of 3 hours, 42 minutes and 24 seconds.  To
qualify for the Boston Marathon Casey needs a time of 3:40:59 (at the regular length.)
```
```a.  If we adjust for the extra length, should Casey qualify for Boston?
b.  Why might a proportional adjustment not be entirely fair?
```
```Notes: From the Cincinnati Enquirer, May 6, 2008.  The paper reports that a multiplier of .9905. They convert
Fein's time to seconds, use the multiplier to convert to a shorter time, and then convert back to
hours, minutes and second to show that Casey does indeed deserve to qualify.
For part b, I hope students will realize that runners will run that added distance at a slower rate
than their average speed - but this is likely the best we can do for an approximation.
```