From the graph above, the number of drops to crack open the nuts depends on the height of the drop. Therefore the independent variable is the height of the drop (h) while the dependent variable is the number of drops (n). The control variable is the size of the nuts which is large for the above specific graph.
Parameters, the characteristic of the population:
The coordinates of the graph form a smooth line. The number of drops is decreasing as the height increases. The height of a drop can never be negative or zero, therefore, h>0. The number of drops will never be negative or less than one, therefore, n≥1.
By estimating the values of parameter, we can infer that the domain and range is the set of strictly positive real numbers.
A function that models the behavior of the graph is a reciprocal function this is because the x axis is a horizontal since f(n) approaches zero from above.
Reciprocal functions have the equation: 1 .
(ax+b)
By use of GDC, geogebra i found the equation by moving the line of the curve to fit accurately. 1 ÷ (0.06x+0.012)-0.08) +1.48
Another function that models the data is the logistic function. This is because it has a horizontal asymptote of 0.
3.6
1-1.55e^ (8-0.31x)
The difference between the reciprocal function and the logistic function is that the logistic function fits more accurately. However, it is still hard to predict other values from the equations
The table below shows the average number of drops it takes to break open a medium nut, and a small nut, from varying heights.
Medium nuts
1÷ (0.02(x-0.04)-(0.02)+0.28)
The logistic equation f (n) for medium nuts= -0.26÷(1-0.99e^0.01x)
Small nuts
1/(0.02x-0.08)
The equation for logistic function f(n) for small nuts is
10.62÷ (1.897e^ (-0.6x)
The first model does not fit the data accurately for nuts of different sizes and it is therefore hard to predict a trend from this. The models were unable to predict undefined values and this were therefore the limitations.
