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. ...read more.
Amount of time for whole tray to go bad [T(in hours)] 3x3 3 4x4 5 5x5 7 6x6 9 7x7 11 From this an expression can be used to descrbe how the size relates to the amount of time, it is 2L - 3 = T . For other tomatoes on the edges and for that matter, any tomato anywhere on a square tray, an expression can be formed using two extra values. I have called these D and D2. D is the distance from the first bad tomato to the side furthest from it, and D2 is the distance from the first bad tomato to the side which is second furthest from it. The expression which shows how long it takes for a square to go bad no matter where the bad tomato is is: D + D2 = T . This works due to the XY rule because the distance A, which is from tomato X to the furthest side from it is the normal route to the furthest tomato from it(going according to the XY rule). And distance B finishes off the route to tomato Y but on the opposite side of the square. The best thing about the above expression is that it works for any tomato going bad first in any sized square. Now, let's have a look at something slightly more complex, an infinitely big tray of tomatoes. If a tomato goes bad in the corner of an infinitely big tray of tomatoes, a triangular number type pattern will happen. I will illustrate it in the table on the next page: : Hour (H) No. bad toms bad afterhour H (N) No. toms going bad at hour H (B) 0 1 1 1 1+2 2 2 1+2+3 3 3 1+2+3+4 4 4 1+2+3+4+5 5 5 etc. These are called triangular numbers because as you write them down they form a triangle type shape. ...read more.
Now if the tomatoes were in a box, stacked in a 3-D form, they would effectively have a Z-axis, aswell as an X and Y to contend with, and infect. If a tomato went bad in the corner of a set of 3 3x3 trays stacked on top of each other, it would take 6 hours for the set to go bad: Hour (H) No. of tomatoes going bad at hour H (B) Total No. of bad toms (N) 0 1 1 1 3 4 2 6 10 3 7 17 4 6 23 5 3 26 6 1 1 Now I'll make a table showing how long it takes other cubed sets of trays to go bad: Size (LxLxL) Time taken for all toms to go bad in hours (T) 2 3 3 6 4 9 5 12 6 15 From this the obvious formula for the amount of time for a cube of tomatoes, or stacked set of trays to go bad is 3L-3 = T . It would be very difficult to try to find any other formulas for cubes of fixed size as specificity of the tomato going bad becomes very complex in 3-D. The last thing which I will look at is an infinitely big 3 dimensional cubed tray with a the first tomato going bad in the middle. It's growth rate pattern will probably be very similar to a 2-D tray, the 2-D pattern was 4H. Here is a table showing howmany tomatoes were going bad each hour, it's on the next page: : Hour (H) No. of toms going bad at hour H (B) 1 6 2 18 3 38 4 66 5 102 The pattern here is in fact very similar to the 2-D infinite square starting at middle one, it's 4N2+2 = B . The squared part of the formula is to add an extra dimension to it. Here Concludeth my Investigation. David Langer 10SM ...read more.
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