The US Census lists the population of the United States as 249 million in 1990, 281 million in 2000, and 309 million in 2010. Fit a second-degree polynomial P(t)=a_{2}t^{2}+a_{1}t+a_{0} passing through these points, where t represents years after 1990 (so t=0 corresponds to 1990) and P(t) represents population in millions (so P(0)=249). Sketch the parabola,P(t). Use the model to predict the population in the years 2020 and 2030. (Source: US Census Bureau). You may use technology to solve the system of 3 equations and 3 unknowns used to find your coefficients/constants for your model. The setup of your 3x3 linear system must be shown.

Three different points are required to determine the coefficients of correspondent second-order polynomial. Three linear equations are form after substituting the variables associated with those points. [tex]t^{*}[/tex] is the year and [tex]p[/tex] is the population according to US census, measured in millions. That is to say:

[tex]a_{2}\cdot 1990^{2} + a_{1}\cdot 1990 + a_{0} = 249\\a_{2}\cdot 2000^{2} + a_{1}\cdot 2000 + a_{0} = 281\\a_{2}\cdot 2010^{2} + a_{1}\cdot 2010 + a_{0} = 309[/tex]

There are different approaches to solve linear equation systems. In this problem, a matrix-based approach will be used and a solver will be applied in order to minimize the effort and time required to make the need operations. The solution of the 3 x 3 linear system is shown as following:

[tex]a_{2} = -\frac{1}{50},a_{1}=83,a_o=-85719[/tex]

Now, the second-order polynomial is:

[tex]p(t)=-\frac{1}{50}\cdot (t+1990)^{2}+83\cdot(t+1990)-85719[/tex], where [tex]p(t) = 249[/tex] when [tex]t=0[/tex].

The predicted populations are:

[tex]p(30) = 333, p(40) = 353[/tex]

US predicted population in 2020 and 2030 will be 333 million and 353 million, respectively.