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?