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Crows Dropping Nuts

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Introduction

Min Hua Ma

9-18-09

IB SL MATHEMATICS

Internal Assessment

Type II

IB SL Type II Internal Assessment: Crows Dropping Nuts

        This assignment is an investigation to find a function that models a given set of data. By using various methods, such as matrixes, different types of regressions, and technology, it allows the investigator/student to create various equations to model the data. This assessment is about birds dropping different sized nuts on a hard surface in a range of heights in order to break open the shells. There are three variables in this investigation: the size of nuts, the heights of drops, and last but not least, the number of drops.

The first set of data is on crows dropping large nuts:

Height of drop

1.7

2

2.9

4.1

5.6

6.3

7

8

10

13.9

Number of drops

42

21

10.3

6.8

5.1

4.8

4.4

4.1

3.7

3.2

To begin this investigation, I began plotting the given points on a scatter plot:

image00.png

Then, I decided to begin with using matrixes to formulate an equation. I wanted to do a matrix using all the points to create this polynomial:

ax9+bx8+ cx7+dx6+ex5+fx4+gx3+hx2+ix+j

I put all the y values in matrix [A] and all the x values in matrix [B].

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Middle

This equation models the points of the last 5 data, but is very close for modeling points 2 and 5.

           Subsequently after using matrices, I realized that using polynomial equations won’t help model the given data since both polynomial equations create a parabola not a decaying model. So I decided that using matrices wasn’t a good way to model the data. I began looking at ways to form to model decay. Using the GCD, I started with using PwrReG.

         To find the formula using PwrReg, press STAT, go to CALC, scroll down to A, and enter again. This will bring the calculator to the main screen with PwrReG, so now press enter again. Then the screen will give the values of “a” and “b”. Plug the information into (y=) and press (GRAPH).

Looking at the graph, this function shows decay and the equation is very close to all of the data, but only goes through the 2nd and 8th point. The regression of the equation is -.95.

         After that, I used the calculator again to find ExpReG. Using the method as the PwrReG, I got the equation:

Y=21.37241705 x (.8358432605x)

Using the given function I plugged in the equation and got:

image14.png

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Conclusion

)-1.2

Small Nuts: y= 37.67398021 x (.894911559191.5x + 2.5) +4

Also, by inferring to the equations above, I see that the medium and small nuts have greater number of drops in order to break open than the large nuts.

                    My first model: Y=20.04386 • (.836011610x) + .5 doesn’t replicate the data of medium and small size nuts because the data from the large nuts had a different heights than both of the small and medium size nuts. The decay equation from the first model is much steeper than the other two. In order to adjust the model to fit the smaller sizes, I first began by making the heights the same, and then compared the ExpReG of each. Even though the heights are the same, the numbers of drops for each of the different sizes of nuts aren’t very similar, so making the first function be able to model the medium and small nuts isn’t very likely. In addition, the regressions of the medium and small nuts aren’t very strong, given that they only model 2 of the 8 points from the data. The limitation in this investigation is: there just can’t be any size for a nut. For example, there can’t be a negative height, some heights aren’t able to break nuts, and the size of nuts must be too large or too small so a seed can’t be 10 pounds.

 Ma

...read more.

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