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Edexcel nov 2011 P2 Q29

Discussion in 'Mathematics' started by Maths_Mike, Feb 22, 2012.

  1. Maths_Mike

    Maths_Mike New commenter

    A rectangle 1m by 50cm has circular disc cut from it. Disc radius is 6cm.

    I gave the answer as 32 which earned the full 2 marks as you can fit 4 rows of 8.

    I undersatnd the area of rectangle shared by area of circle gives 44 (which is wrong) but was apparently worthy of one mark.
    What I don't understand is why the correct answer was given as 36 and why any integer value inbetween 32 and 36 was accepted fro full marks.
    I am guessing it might be possible to arrange the circles in such a way that 36 will fit but have no idea how to show this and cant believe it would be expected from a foundation student!!
     
  2. arsinh

    arsinh New commenter

    Fitting 36 discs onto this rectangle is anything but trivial. The Examiner's Report says:
    "The optimal answer of 36 was very rarely seen."
    I'm not surprised!

     
  3. Maths_Mike

    Maths_Mike New commenter

    well it wasnt seen once when I possed the questio in the maths office today despite being done by experienced teachers of maths / a Level / furhter maths etc. I for one admit to still not knowing how to get the answer of 36.
    anyone care to enlighten me?
     
  4. DM

    DM New commenter

    I haven't tried to get more than 32 (4 rows of 8 is the obvious answer but it leaves the sneaking suspicion that more might be possible).
    Does anyone have twiddlywinks or loads of other flat circular counters at home? The quickest way to find the solution might be to tip 36 on the floor, draw a rectangle to scale based on their diameter and jiggle them about a bit!
     
  5. Maths_Mike

    Maths_Mike New commenter

    Just found a document suggest the optmal value is 81% coverage (which works for 36)
    When the chief examiner said the optimal answer was rarely seen I think he was telling a Porky Pie because short of a lucky guess I cant believe anyone would have got this.
     
  6. I am sure that the expected answer was 4x8 but since this is not the maximum number anyone putting 33-36 would still have to be awarded the same marks as 32 (33 being just as incorrect as 32). The correct answer of 36 is obtained by cutting out 5 equally spaced circles from the base of the rectangle ( with one in each corner) then cutting out a row of 4 sitting in the gaps between the five on the base. You can then repeat this combination 4 times which gives 4 rows of 9. This gives the same design pattern used in packing oranges into a rectangular tray, check it out next time you are in sainsburys!

    Not quite sure this would be a foundation paper question though, might be something my A level students would like to prove. However, I believe it was recently proved that the best way to stack oranges in a 3-d arrangement was a pyramid shape, something the local green grocer has known for years yet only recently proved using some very advanced topology!
     
  7. Maths_Mike

    Maths_Mike New commenter

    5 circles would have gaps of 10cm in between. The circle in the middle would need to have its centre 4.79cm higher giving a total width across a row of 9 as 16.79, thus you could not get 4 lots of this arrangement across the 50 cm width (if my calculation are correct)

    Edexcel are certainly making the foundation exams harder these days!!
     
  8. so the height of its top is 16.79..

    If you put three more circles vertically above this one the top of the top one is at 36+16.79..=52.79..

    Have I misunderstood your proposed arrangement?
     
  9. Maths_Mike

    Maths_Mike New commenter

    Yes sry I did 12 + 8 SQRT 23 and concluded it would not fit, but actually I need We only need 7 SQRT 23 for the gaps + 6 at each end giving 45.5 so fits with room to spare.

    Feeling dumb that I couldn't do a foundation gcse question!

    [​IMG]
     
  10. Yes,
    row of 5 on the bottom, all equally spaced, then put 4 in the gaps, then start again with 5 on the next row then 4, then 5 followed by 4 etc.
    [​IMG]
    Similar to this design, but with rows of 4 and 5.
     
  11. or even better

    [​IMG]


     
  12. DM

    DM New commenter

    Sorry but these proposed "solutions" sound like drivel to me.
    Sets of 5 with an offset of 4 will produce this result which clearly doesn't fit:
    [​IMG]
    Here is my students' life-sized effort (sorry about their photography skills). I am suggesting the "optimal" answer is actually 34.

    [​IMG]

     
  13. GoldMaths

    GoldMaths New commenter

    Agreed best I managed with my A-Level Further Maths class was 34....
     
  14. DM

    DM New commenter

    On further inspection, I have found that 34 results in a tiny overlap. 33 looks best after all.
     
  15. Maths_Mike

    Maths_Mike New commenter

    OMG this is driving me nuts!
    I originally couldnt get 36 and didnt believe the 5/4 array until someone convinced me otherwise so glad DM has proved this doesnt work.
    Also when I suggested some irregular pattern at work I was shot down in flames as an "irregular pattern cant possible be optimal" - or at least that's what I was told!
    Any I am now desperate to nkow if and how 36 (the optimal solution accorcing to Edexcel) can be achieved ! - any university professoes want to have ago at this GCSE grade C question - poecat perhaps or subject knowledge !
     
  16. DM

    DM New commenter

    33 is fine. 34 slightly elliptical CDs is possible. 35 and 36 - do me a favour!
     
  17. coyote

    coyote New commenter

    See http://en.wikipedia.org/wiki/Circle_packing_in_a_square on this - sometimes the optimal solution is irregular! Or at least, not as symmetrical as you might think.
    I've also found this website http://www.packomania.com/ which suggests that if you want to fit in 36 circles, the maximum radius is just over 5.8 cm, and shows an arrangement of 34 circles radius just under 6cm fitting. And it's irregular!
     
  18. DM

    DM New commenter

    Fab site coyote - nice to know that my calculations about 34 not quite working were correct. There is still symmetry in the 34 arrangement.
     
  19. Maths_Mike

    Maths_Mike New commenter

    Thanks - my mind is at ease and we conclude yet another edexcel mistake - just not one that made the news.
     
  20. As far as I was aware, there is no known way of knowing the optimal solution for packing circles but, I think it has been shown that given an infinte space to work in a hexagonal arrangement gives the best coverage. Can't remember where this came from so I might just have made it all up!
     

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