How do you come up with mathematical models?

[magicmath]

p = 4x/5 + 25

x is the number of years after 1980 and p is the percentageof U.S. college freshmen who had an average grade of A in highschool.

With this mathematical model, we can predict the future.

My question is

How did you come up with this formula in the first place to do this?

What's the procedure?

 

[Dr.Peterson]

p = 4x/5 + 25

x is the number of years after 1980 and p is the percentageof U.S. college freshmen who had an average grade of A in highschool.

With this mathematical model, we can predict the future.

My question is

How did you come up with this formula in the first place to do this?

What's the procedure?

In this case, they just made it up -- it's a fake, used just for practice. It doesn't make sense as a real model, unless they at least state a time period over which it would apply. (What is p in the year 2080?)

If this model were real, they might have observed data showing approximately linear growth over some time interval and used linear regression to determine the equation of a line matching the graph. Then they would say that this approximately describes the data for that time period, and can only be extrapolated to later times with great care. ("Past performance does not necessarily predict future results".)