# Bad Tomatoes

Extracts from this document...

Introduction

Natalie Hayes 10E group 3Page 1

## Bad Tomatoes

I have been given a box filled with identical good tomatoes, one of the tomatoes goes bad and after 1 hour, all the tomatoes that it touches go bad. After another hour, the bad tomatoes make all the good tomatoes they touch go bad. This continues until all the tomatoes in the box go bad.

My aim in this investigation is to find a formula to show how many tomatoes in a tray go bad.

I have started the investigation by drawing diagrams to help get me started, I thought it would be more useful if I investigated how

Middle

As you can see from the diagrams that the amount of tomatoes that go bad depends on the position of the first bad tomato. If I was going to investigate the amount of tomatoes that would go bad win a tray with unlimited sides then I would have to draw myself bigger diagrams to help me find a sequence (look at page 6)

### Corner

Here is a table to show the total amount of bad tomatoes after a certain amount of hours, with the bad tomato starting in the corner in a box with unlimited sides.

Hour | ## Total number of bad tomatoes |

0 | 1 |

1 | 3 |

2 | 6 |

3 | 10 |

4 | 15 |

5 | 21 |

By looking at this table of results, I can now produce a sequence.

1 2 3 4 5

3 6 10 15 21

+3 +4 +5 +6

+1 +1 +1

## Natalie Hayes 10E group 3Page 2

The first thing I noticed when looking at my sequence was that the difference between each number increased by 1 each time e.g. +1, +2, +3, +4, +5, +6.

As you can see from the sequence, second differences are present. I used the formula An² + Bn + C to help me work out an nth term. The first thing I did was to work out the simultaneous equations as I knew that it would find what A, B an C represented.

- A + B + C = 3
- 4A + 2B + C = 6
- 9A + 3B + C = 10
- 16A + 4B + C = 15

the next stage was to

Conclusion

Box sides | Number of hours for tomatoes to go bad |

3 by 3 | 3 |

4 by 4 | 5 |

5 by 5 | 7 |

6 by 6 | 9 |

3 4 5 6

3 5 7 9

+2 +2 +2

There is also 1st differences present so again I will use simultaneous equations to work out a formula to find the nth term.

1. A + B = 3

2. 2A + B = 5

3. 3A + B = 7

- 4A + B = 9

- 2A + B = 5

- 1. A + B = 3

##### A = 2

Natalie Hayes 10E Group 3Page 11

If I substitute 2 for A in equation 2. then I will get 4 + B = 5

Therefore B = 1

Nth term = 2n + 1

## Corner

When investigating if the figures change in different size trays when the starting bad tomato is positioned in the middle I realised that I could only use trays with odd sides e.g. 3 by 3, 5 by 5. as trays with even sides do not have an exact centre position. Here are my diagrams that I used and a table to show my results.

Box size | Number of hours for tomatoes to go bad |

3 | 2 |

5 | 4 |

7 | 6 |

9 | 8 |

Natalie Hayes 10E Group 3Page 12

3 5 7 9

2 4 6 8

+2 +2 +2

I can see by looking at my results that the nth term is n - 1, for example it would take 10 hours for the tomatoes to go bad in a box of 11 by 11.

## Conclusion

For part 1 of this investigation I have found out how the tomatoes go bad in trays of different sizes and trays of unlimited sides, I have noticed differences and patterns if you change the position of the starting bad tomato. After researching and using my knowledge I have been able to find many formula’s so it is now possible to understand how the tomatoes in the tray go bad.

This student written piece of work is one of many that can be found in our GCSE Bad Tomatoes section.

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