Question

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.

Answer 1

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

Step-by-step explanation:

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. $t^{*}$ is the year and $p$ is the population according to US census, measured in millions. That is to say:

$a_{2}\cdot 1990^{2} + a_{1}\cdot 1990 + a_{0} = 249\\a_{2}\cdot 2000^{2} + a_{1}\cdot 2000 + a_{0} = 281\\a_{2}\cdot 2010^{2} + a_{1}\cdot 2010 + a_{0} = 309$

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:

$a_{2} = -\frac{1}{50},a_{1}=83,a_o=-85719$

Now, the second-order polynomial is:

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

The predicted populations are:

$p(30) = 333, p(40) = 353$

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