The heptathlon is an athletics competition consisting of 7 events:

• 200m sprint
• 800m run
• 100m hurdles
• high jump
• long jump
• shot put
• javelin
The scoring system uses two types of equation:

$y=a(b-x)^c$        (1)
$y=a(x-b)^c$        (2)

$a$, $b$ and $c$ are constants, $x$ is the competitor's time or distance and $y$ is the number of points they are awarded.

Which events do you think use equations of type (1)? Why?
Which events do you think use equations of type (2)? Why?

For the running events, $x$ is the time in seconds. For the jumping events, $x$ is the distance/height in centimetres. For the throwing events, $x$ is the distance in metres.

The values for a, b, and c in each event are given below:

 Event a b c 200 meters 4.99087 42.5 1.81 800 meters 0.11193 254 1.88 100 metres hurdles 9.23076 26.7 1.835 High Jump 1.84523 75 1.348 Long Jump 0.188807 210 1.41 Shot Put 56.0211 1.5 1.05 Javelin Throw 15.9803 3.8 1.04

In the table below are the best times and distances of an Olympic hopeful in training, as well as the World Records for each heptathlon event (as of April 2011).

 Event Olympic hopeful World records 200m 25.34s 21.34s 800m 2min 13.00s 1min 53.28s 100m hurdles 13.65s 12.21s High jump 1.43m 2.09m Long jump 5.67m 7.52m Shot put 12.45m 22.63m Javelin 45.05m 72.28m

In order to work out a suitable training schedule for her, work out her score in each event.

The world record for each event can be taken as a theoretical maximum/minimum.
Suppose she could close the gap between her current performance in each event and the world record by 10%. How would that affect her progress towards her target heptathlon score of 6000 points?

Instead, she could put together an alternative training schedule aiming to close the gap by 20% in some events. However, this extra training would have to be at the expense of her training for other events (so for every event she chooses to improve by 20%, she must choose another where she forfeits the 10% gain and instead maintains her current level).

Could this training strategy lead to a better score?
Can she reach the target of 6000 points?

NOTES AND BACKGROUND

Data source: http://en.wikipedia.org/wiki/Heptathlon