Answer
See Below
Work Step by Step
(a)
The quadratic equation is:
$p=0.004{{x}^{2}}-0.36x+14$
It is required to calculate percentage of U.S population that was foreign born in 1990.
The value of \[x\] is:
\[\begin{align}
& x=1990-1920 \\
& =70
\end{align}\]
Now, the percentage of U.S population can be calculated as:
$\begin{align}
& p=0.004{{\left( 70 \right)}^{2}}-0.36\left( 70 \right)+14 \\
& =8.4
\end{align}$
According to the graphical data, the percentage of population that was foreign born in 2000 is 10.4.
So, the model is overestimate. The difference is:
\[8.4-8=0.4\]
This overestimates the actual number by 0.4%.
The provided model overestimates by 0.4%.
(b)
Percentage of U.S population is 23%. So, \[p=23\].
Therefore:
$\begin{align}
& p=0.004{{x}^{2}}-0.36x+14 \\
& 23=0.004{{x}^{2}}-0.36x+14 \\
& 0.004{{x}^{2}}-0.36x-9=0
\end{align}$
Here, quadratic equation with coefficients $a=0.004,b=-0.36\text{ and }c=-9$.
So:
$\begin{align}
& x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a} \\
& =\frac{-0.36\pm \sqrt{{{0.36}^{2}}-4\left( 0.004 \right)\left( -9 \right)}}{2\left( 0.004 \right)} \\
& \approx -20,110
\end{align}$
Number of years cannot be negative. So, \[x=110\] is considered.
Now, the year can be calculated as:
\[1920+110=2030\]
Hence, the year to reach population to 23 percent is 2030.
