Good Ol' Seagulls and Clams - Creating an equation based on data

[soccergirl14]

So pretty much, I was given data (see below) about seagulls dropping clams and the number of drops required to break the clam.

Height of drops (x)/number of drops (y)
1.7 / 42
2 / 21
2.9 / 10.3
4.1 / 6.8
5.6 /5.1
6.3 / 4.8
7 /4.4
8 /4.1
10 /3.7
13.9 /3.2



So for the last six points I created the equation y= 1/ (0.045x) +1 which showed there was a horizontal asymptote of 1 (because you can't have less than one drop), a vertical asymptote of 0 (because you can't drop the clam from a distance from less than zero), and no x-intercepts. I'm trying to create a piece wise function, but I'm having trouble getting the first point to relate to the other points in an equation.

 

[diazona]

What problem exactly are you trying to solve?

 

[Architeuthis Dux]

I would first commend you for using data to create an equation rather than just assuming a relationship based on intuition. It is always important to have evidence to support our conclusions.

Based on the data provided, it seems that there is a clear inverse relationship between the height of the drop and the number of drops required to break the clam. This means that as the height of the drop increases, the number of drops required decreases.

To create an equation that represents this relationship, we can use the concept of inverse variation. The equation would be y = k/x, where k is a constant. To find the value of k, we can use any of the data points given. Let's use the first data point (1.7, 42). Substituting these values into the equation, we get 42 = k/1.7. Solving for k, we get k = 71.4.

Therefore, the equation that represents the relationship between the height of the drop and the number of drops required to break the clam is y = 71.4/x. This equation also satisfies the conditions of a horizontal asymptote of 1 and a vertical asymptote of 0.

To create a piecewise function, we can use the equation y = 71.4/x for all values of x greater than 0. For x = 0, we can use the value of 1 as the number of drops required cannot be less than 1. Therefore, the piecewise function would be:

y = 71.4/x, for x > 0
y = 1, for x = 0

This function accurately represents the data provided and can be used to predict the number of drops required to break a clam at any given height. However, it is important to note that this equation is based on the data provided and may not be applicable to all situations involving seagulls and clams. Further research and data collection may be necessary to validate this equation.

In conclusion, using data to create an equation is a valuable approach in scientific research. It allows us to make accurate predictions and understand the relationship between variables. However, it is important to continuously test and validate our equations to ensure their applicability in different scenarios.