GCSE Maths Bad Tomato Investigation

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GCSE Maths Bad Tomato Investigation

GCSE Maths Bad Tomato Investigation

 

A problem has emerged involving bad tomatoes. I am going to investigate it and find some solutions to the problem. At first there is a regular tray of good tomatoes. Then suddenly, one goes bad. An hour later, all the ones that the bad tomato is touching go bad. This continues until the whole tray goes bad. The problem gets worse when the trays are stacked on top of each other, but I’ll look at that later.

 

I’m going to start by investigating a normal square box of tomatoes. It would look like this to start with:

                                                                                                                                                                                                                                           

 

                                                                                                                                                                                                                                   

 

 

 

 

Then let’s say tomato 1 goes bad:

 

 

 

 

 

 

 

 

 


After one hour, tomatoes 1, 2 and 5 will be bad:

 

 

 


                                                                                                                          

 

 

 

 

 

Then after another hour tomatoes 3, 6 and 9 will go bad. This will continue until the whole tray of tomatoes is bad:

 

 

 

 

 

 

 

 

 


After looking at this process, a pattern became obvious. This was that the quickest legal route to the furthest tomato away from the original bad tomato would be the number of hours for the whole tray to go bad. Basically, if tomato X goes bad, then the quickest legal route to tomato Y, being the furthest tomato away from tomato X would be the number of hours for the whole tray to go bad. Let’s call this the XY rule. This 4 tomato by 4 tomato tray would take 6 hours to go bad:

 

 

 

 

 

 

 

 


The order of tomatoes going bad is illustrated in this table:

 


Hour (H)              No. of bad toms after that hour (N)               Tomatoes which are bad

 


0                                        1                                                                                     1

1                                        3                                                                                 1, 2, 5.

2                                        5                                                                           1, 2, 5, 3, 6, 9.

3                                        9                                                        1, 2, 5, 3, 6, 9, 4, 7, 10, 13.

4                                        12                                       1, 2, 5, 3, 6, 9, 4, 7, 10, 13, 8, 11, 14.

5                                        15                          1, 2, 5, 3, 6, 9, 4, 7, 10, 13, 8, 11, 14, 12, 15.

6                                              16                        1, 2, 5, 3, 6, 9, 4, 7, 10, 13, 8, 11, 14, 12, 15, 16.  

 

 

A clear pattern can be seen when looking at square trays with a tomato going bad in the corner. Here is a table showing some different sized squares and how long they take to go bad with the bad tomato starting off in the corner:

 


Size (LxL)                       Amount of time for whole tray to go bad [T(in hours)]

 


2x2                                                 2

3x3                                                 4

4x4                                                 6

5x5                                                 8

6x6                                                 10

7x7                                                 12

 

The obvious pattern is that as length - l increases by one, the time for the tray to go bad - t increases by 2. This is because using the XY rule, the amount of time to get from corner to opposite corner will always increase by two as l increases by 1. This in turn is because there will be an extra tomato on each of the two sides on route from corner to corner, there by increasing the time for the tray to go bad by 2 hours. This is shown in the diagram on the next page:

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After examining all of this information, an expression can be used to describe it:

2L-2 = T This means that twice the length of side - L minus 2) will equal the time for the whole tray to go bad. The reason for this expression working is using the XY rule, when a tomato in the corner goes bad, the one in the opposite corner will be the furthest one away. In this expression the two sides on route from corner to corner are represented by 2L but there is ...

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