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Crows dropping nuts

Extracts from this document...

Introduction

Mathematics Coursework 2009: Crows Dropping Nuts

Standard Level

Introduction

The following table shows the average number of drops it takes to break open a large nut from different heights.

Large Nuts

Height of drop in meters (𝒙)

1.7

2.0

2.9

4.1

5.6

6.3

7.0

8.0

10.0

13.9

Number of drops (𝒚)

42.0

21.0

10.3

6.8

5.1

4.8

4.4

4.1

3.7

3.2

The following line graph, portraying the above table, shows us the number of drops by the height for each drop.

image00.png

Variables:

In this graph, there many different variables that we need to perceive. One variable is how big the nut is. How large the nut is will invariably affect the number of times that the nut is dropped until it cracks open. By stating that the nuts are different sizes, it will help us recognize the nature of the model. Another variable is the fact that the number of drops are described as an average, because it is impossible (for example) to drop something 10.3 times. However, the reason that it has been recorded as an average is because it gives us more lucid data, which can be transcribed as a graph more easily. An additional variable is how the height that the nut is dropped affects the number of times the nut is dropped, and how it is put into an average.

image00.png

Parameters:

A parameter that can be stated is that in the above graph, there are asymptotes on both the 𝒙 and 𝒚 axes. This means that this graph isn’t exact, where there can be an unlimited number of possibilities for the values on the axes.

...read more.

Middle

image14.png

My model does not have as steep a descent of that of the original ‘large nut’ graph and the curve of my graph drops to just below that of the original’s, but then equalizes with the original at the point ,𝟏𝟑.𝟗,𝟑.𝟐.. In my model, the asymptote matches that of the original’s exactly on the 𝒙−𝒂𝒙𝒊𝒔, but not on the 𝒚−𝒂𝒙𝒊𝒔. My model crosses on 3 points of that on the original; ,𝟏.𝟕,𝟒𝟐., ,𝟒.𝟏,𝟔.𝟖.,  and ,𝟏𝟑.𝟗,𝟑.𝟐.. Though there are some similarities, the results are not particularly satisfying. So, we will use the trial and error method with 𝒚=,𝒙-−𝟏.which will result in a more accurate answer, because since we will be using more parameters, we will be able to shape the equation more closely to the original ‘large nut’ graph.

𝒚=,𝒙-−𝟏.

By using the method of trial and error, we will estimate an approximately accurate model to the original ‘large nut’  graph. We will manipulate the function by using the following format:

𝒚=,𝒄(𝒙+𝒅)-−𝟏.+𝒆

 𝒚=,𝒙-−𝟏.

image15.png

We will increase the value of 𝒄in 𝒚=,𝒄(𝒙+𝒅)-−𝟏.+𝒆

 𝒚=𝟐𝟎(,𝒙-−𝟏.)

image16.png

We will increase the value of 𝒄in 𝒚=,𝒄(𝒙+𝒅)-−𝟏.+𝒆

𝒚=𝟐𝟖(,𝒙-−𝟏.)

We will increase the value of𝒄, and decrease the value of 𝒅 in 𝒚=,𝒄(𝒙+𝒅)-−𝟏.+𝒆image17.png

𝒚=𝟐𝟖(,𝒙−𝟏.𝟎𝟒)-−𝟏.

image18.png

We will decrease the values of 𝒄and 𝒅 and increase the value of 𝒆 in 𝒚=,𝒄(𝒙+𝒅)-−𝟏.+𝒆

𝒚=𝟐𝟏.𝟕(,𝒙−𝟏.𝟏𝟑)-−𝟏.+𝟏.𝟒𝟓

image01.png

We will decrease the value of𝒄 and increase the values of 𝒅 and 𝒆 in 𝒚=,𝒄(𝒙+𝒅)-−𝟏.+𝒆

𝒚=𝟏𝟖.𝟕(,𝒙−.𝟑𝟕)-−𝟏.+𝟏.𝟕𝟖

image02.png

We will decrease the values of𝒄 and 𝒆 increase the value of 𝒅 in 𝒚=,𝒄(𝒙+𝒅)-−𝟏.+𝒆

𝒚=𝟏𝟐.𝟕(,𝒙−𝟏.𝟑𝟖𝟒)-−𝟏.+𝟏.𝟕𝟔

image03.png

We will increase the values of𝒄 and 𝒆

...read more.

Conclusion

   -

57.0

19.0

14.7

12.3

9.7

13.3

9.5

Graph showing the medium nuts (red line) versus 𝒚=𝟏𝟐.𝟗𝟏(,𝒙−𝟏.𝟑𝟕)-−𝟏.+𝟐.𝟐𝟏

(blue line)

image09.png

We will manipulate the graph (via the trial and error method) by manipulating the function in the 𝒚=,𝒄(𝒙+𝒅)-−𝟏.+𝒆format.

We need to increase the values of 𝒄 and 𝒆 and decrease the value of 𝒅 in 𝒚=,𝒄(𝒙+𝒅)-−𝟏.+𝒆

𝒚=𝟐𝟏.𝟔𝟓(,𝒙−𝟑.𝟓𝟕)-−𝟏.+𝟔.𝟗

image10.png

We are only able to get the general shape because of the structure of the original data points, which does not make very accurate. The limitations of the model is that it is done by trial and error. So it will never be exact no matter what we do to improve it. Also, because of the irregularity of the line graphs of the small and medium nut graphs, the models will never be able to fully replicate them. The medium and small nut graphs are not real curves, and because the models that have been created will always be curves, it will be impossible to create an ideal graph.

The function 𝒚=𝟏𝟐.𝟗𝟏(,𝒙−𝟏.𝟑𝟕)-−𝟏.+𝟐.𝟐𝟏 does not particularily depict the medium and small nut graphs very well. However, by using the method of trial and error, we have been able to manipulate the graphs into the general shape of the original graphs. As said earlier, an ideal graph will never be created, but by this method the slope, area under the curve, and other properties will be similar to that of the originals’. A limitation is that the graph will continue almost straight up. In reality and in the medium and small nut graphs, as the height of the drop (𝒚−𝒂𝒙𝒊𝒔) decreases so will the number of drops (𝒙−𝒂𝒙𝒊𝒔). In this case, the models will never work.

...read more.

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