Problem:
Given the following table:

Horizontal
Distance
Release
Height
5731000
534800
495600
451450
395300
337200
253100
[Extracted from Galileo's Gravity and Motion Experiments.]

find and graph the third degree polynomial model which best approximates the data.


Visualization:
Using Maple:

First, we'll plot the data points:

> with(plots):
> plot([[573,1000], [534,800], [495,600], [451,450], [395,300], [337,200], [253,100]], style=point, symbol=diamond, color=blue);

We'll store this information using the name dataplot:

> dataplot:=":

We'll now use the Maple command fit:

> with(stats):
> fit[leastsquare[[x,y], y=a*x^3+b*x^2+c*x+d]]([[573, 534, 495, 451, 395, 337, 253], [1000, 800, 600, 450, 300, 200, 100]]);

We now simplify the polynomial by using real numbers instead of fractions:

> evalf(14434878262330546175/1041410474157670475856608*x^3 - 4682342194290644730025/520705237078835237928304*x^2 + 2921879928533310334736425/1041410474157670475856608*x - 134268118430574577219730675/520705237078835237928304,5);

We now plot the graph of this polynomial:

> plot(.13861e-4*x^3-.89923e-2*x^2+2.8057*x-257.86, x=225..575);

We'll store this information using the name fctplot:

> fctplot:=":

Finally, we combine these plots:

> display({dataplot,fctplot}, view=[225..575,75..1000]);