• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month
  1. 1
  2. 2
  3. 3
  4. 4
  5. 5
  6. 6
  7. 7
  8. 8
  9. 9
  10. 10
  11. 11
  12. 12
  13. 13
  14. 14
  15. 15

Crows dropping nuts

Extracts from this document...


Mathematics Coursework 2009: Crows Dropping Nuts

Standard Level


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)











Number of drops (𝒚)











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



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.



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.



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.


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:




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



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


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



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



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



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



We will increase the values of𝒄 and 𝒆

...read more.










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

(blue line)


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



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.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related International Baccalaureate Maths essays

  1. Crows dropping nuts

    Therefore it is expected these differences from trial to trial would account for variation in any model attempting to fit the data, as these would have significant contributions towards the nut cracking from different heights. As well as it was also not stated if the craw was stationary when dropping

  2. A logistic model

    The interval of calculation is 1 year. Year Population Year Population 1 1.0?104 11 7.07?104 2 2.9?104 12 4.19?104 3 6.32?104 13 7.07?104 4 5.56?104 14 4.19?104 5 6.49?104 15 7.07?104 6 5.28?104 16 4.19?104 7 6.73?104 17 7.07?104 8 4.87?104 18 4.19?104 9 6.96?104 19 7.07?104 10 4.42?104 20 4.19?104 Estimated magnitude of population of fish

  1. Maths Modelling. Crows love nuts but their beaks are not strong enough to ...

    In the above function "c" is used to represent the horizontal shift and "d" the vertical shift. To find the values for a, b, c and d I looked at the domain and range and this helped me find the values for c and d.

  2. Math portfolio: Modeling a functional building The task is to design a roof ...

    to "v" and then equate to zero =150[(1296v - v3)]=0 Therefore 1296-3v2=0 -3V2=-1296 V=�20.78m This value can be maxima or minima hence I will differentiate the equation (8) again and check whether the result is positive or negative. =(-6V)=-124.68 As the above value is negative, hence the value of "v"20.78 is for largest cuboid.

  1. Crows Dropping Nuts

    Next, I used the last 5 data points from the data to create an equation. 6.3 7 8 10 13.9 4.8 4.4 4.1 3.7 3.2 Using the same method as the first 5 data points, I created the following

  2. Creating a logistic model

    + b 1 = m(60000) + b Solving this on the GDC, we have m = -0.00002 and b = 2.2 so the linear growth factor is: rn = -0.00002 � un + 2.2 and therefore the function for un+1 for is: un+1 = (-0.00002 � un + 2.2)

  1. Investigating the Graphs of Sine Functions.

    Graph of y= -2 sin x Conjecture: (a) transformation of the standard curve y= sin x: from the examples above I can deduce that the amplitude changes according to the coefficient, therefore y= -2sin x must be a dilation along the y-axis by the factor of two. Because the coefficient is now negative I conjecture that the graph

  2. How many drops does it take for a crow to crack a nut?

    = 49.9x^-1.25 where x = Height of Drop (m); x > 0 and y = Number of Drops The parameters are a and b. The choice of ordered pairs was randomly selected. The line f(x) does not go through all points, but it does go through some making it a line of best fit.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work