# Crows Dropping Nuts

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:

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

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].Then I took the inverse of matrix [B], multiplied by matrix [A], in order to find the values of each letter in matrix [C]:

[B] x [C] = [A]

[C] = [B]-1 x [A]

But, this method was unable to solve for what [C] was because there isn’t an inverse for all of the numbers, meaning there is no inverse for matrix [B] and therefore this way of creating a polynomial equation using all of the data does not work.

Knowing that I couldn’t make a polynomial from all 10 of the data points, I decided to split the data points to the first 5 points and the second 5 points. The first 5 points are:

[A]= and [B]=

so, [C] = [B]-1 x [A]

[C]=

= 4.46781392a4+ -66.20812432b3+353.927455c2+-812.134389d+687.7474501. I graphed the equation using a GDC:

This function modeled all of the first 5 data points, but this equation doesn’t model the scatter plot’s decreasing values.

Next, I used the last 5 data points from the data to create an equation.

Using the same method as the first 5 data points, I created the following equation:

y= .0041005051a4+ -.1625963516b3+ 2.380130469c2 + -15.47352983d + 42.01355157

, and entered the equation to graph:

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

Subsequently after using ...