Answer
SHOWN BELOW
Work Step by Step
(a)
The quadratic equation is:
$p=0.004{{x}^{2}}-0.36x+14$
Since, it is required to calculate percentage of U.S population was foreign born in 2000.
So, the value of \[x\] is:
\[\begin{align}
& x=2000-1920 \\
& =80
\end{align}\]
Now, the percentage of U.S population can be calculated as:
$\begin{align}
& p=0.004{{\left( 80 \right)}^{2}}-0.36\left( 80 \right)+14 \\
& =10.8
\end{align}$
According to graphical data percentage of population, foreign born in 2000 is 10.4.
So, the model is overestimate. The difference is:
\[10.8-10.4=0.4\]
This overestimates the actual number by 0.4%.
The provided model overestimates by 0.4%.
(b)
Percentage of U.S population is 18%.So, \[p=18\].
Therefore:
$\begin{align}
& p=0.004{{x}^{2}}-0.36x+14 \\
& 18=0.004{{x}^{2}}-0.36x+14 \\
& 0.004{{x}^{2}}-0.36x-4=0
\end{align}$
Here, quadratic equation is with the coefficients $a=0.004,b=-0.36\text{ and }c=-4$.
So,
$\begin{align}
& x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a} \\
& x=\frac{-0.36\pm \sqrt{{{0.36}^{2}}-4\left( 0.004 \right)\left( -4 \right)}}{2\left( 0.004 \right)} \\
& x=100,-10 \\
\end{align}$
Number of years cannot be negative. So, \[x=100\]is considered.
Now, the year can be calculated as:
\[1920+100=2020\]
Hence, the year to reach population to 18% is 2020.
