# Crows dropping nuts

Introduction

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

Large Nuts

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

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.

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. In the graph, when the height of the drops (𝒙−𝒂𝒙𝒊𝒔) is too low, the number of drops (𝒚−𝒂𝒙𝒊𝒔) is too high. And in reality, this would be impossible, so we must propose parameters for this graph. This means that the graph cannot touch the axes, because the model would not work. For example, if the height of the drops was 0 meters, then there would have to be 𝒚 number of drops, which is physically impossible.

One type of function that can model the behavior of the graph can be the inverse exponential 𝒚=,𝒆-−𝒙.

Another type of function that can model the behavior of the graph is 𝒚=,𝒙-−𝟏.

By just observing these two graphs, the inverse exponential looks as if it can be manipulated in such a way that it can represent the original ‘large nut’ graph.

When looking at all three functions on one graph, we see the following:

In order to find the equation of the inverse exponential graph 𝒚=,𝒆-−𝒙., we will use simultaneous equations. And for 𝒚=,𝒙-−𝟏., we will use the autograph system via trial and error.

𝒚=,𝒆-−𝒙.

To create a model for the inverse exponential graph, we can use simultaneous equations. We need to create two different constants, and use information from the table to ascertain the equation. So, by using 𝒚=𝒂,𝒆-−𝒙.+𝒃 and finding 𝒂 and 𝒃 (the two constants), we should be able to form a fairly accurate ...