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Maths Modelling. Crows love nuts but their beaks are not strong enough to break some nuts open. To crack open the shells, they will repeatedly drop the nut on a hard surface until it opens. So, through this portfolio I will attempt to create a function t

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Introduction

Haldar

Crows Dropping Nuts

In this portfolio we are given some interesting information about crows. Crows love nuts but their beaks are not strong enough to break some nuts open. To crack open the shells, they will repeatedly drop the nut on a hard surface until it opens. So, through this portfolio I will attempt to create a function that models this behaviour of the crows. Below is a table that shows this data.

This table shows the average number of drops it takes to break open a large nut from various heights

Large Nuts

Height of drop (m)

1.7

2.0

2.9

4.1

5.6

6.3

7.0

8.0

10.0

13.9

Number of drops

42.0

21.0

10.3

6.8

5.1

4.8

4.4

4.1

3.7

3.2

Variables:

There are some variables in this graph. The first variables that I am going to define are the independent and dependant variables. The independent variable that I chose was the height of drop which I will represent with (x). The dependant variable that I chose was the number of drops which I will represent with (y). Thus, all graphs will be shown as (x) – height of drop and (y) – number of drops. Another variable that will be accounted for in the portfolio is the size of the nuts which will be represented with the variables (sn) – small nuts or (mn) – medium nuts or (ln) – large nuts.

Parameters:

After analyzing the data given I noticed some constraints in the data given. Firstly it is important to note that the data given is an average of the number of drops taken to break open a large nut.

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Middle

image08.png

I feel that this function fits the data points much better because it goes through most of the points and touches all of the other ones. The domain for this function is D:image09.png and the range of this function would be R:image10.png. The domain and range of this function has been modified to from the original domain and range to find a better curve of best fit. I feel that by modifying the domain and range I was able to find a better curve of best fit as well as meet all the parameters outlined above.

To compare my equation to another equation, I had to choose which regression to use. After looking at the list I immediately ruled out the linear, the quadratic as well as the sine regression because it was highly unlikely that they would provide a curve of best fit for the data points. This left me with only two options which were the exponential regression and the power regression. So, I decided to try out both of them to see which fit the points the best. Below is a screenshot showing the equations of the exponential as well as the power regression.

image11.pngimage13.png

By comparing the r2 values of the two it is quite clear to see that the power regression is a better fit for the points than the exponential regression. Thus, it makes sense to compare my function with the power regression function to see which is a better fit for the data points.

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Conclusion

image32.png

Below is a graph comparing the equation that I found and the equation I found using power regression.

image33.png

The function that I found was (sn) => y =image34.png , which is represented by f(x) in the above graph. The function that I found using the power regression was                         (sn) => y =137.13(x)-1.09, which is represented by g(x) in the above graph. From this graph it is clear to see that the equation that I found is much better than the equation found using the power regression. I think this occurs because the r2 value of the power regression is 0.618 which is not at all close to one and this does not make it a good curve of best fit. The function that I found also touches more points. Whereas, the function found using the power regression does not touch any points.  Thus, the function that I found fits better than the function found by the calculator.

Therefore, through this portfolio I learnt a lot about functions and how they are used to model a set of data. I also learnt that many number of functions can be used to model a particular set of data as long as you restrict its domain and range. Through this portfolio I was also able to understand a lot about regressions and how they work and that the r2 value of a regression can tell us a lot about the function. By completing this portfolio I was able to understand a lot about functions.  

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