Answer
a. $\frac{P(x)}{Q(x)}=\frac{$1.75x^2-15.9x+160}{2.1x^2-3.5x+296}$.
b. $68\%$
c. $66\%$
d. $y=0.83$
Work Step by Step
a. the human resource budget is $P(x)=1.75x^2-15.9x+160$, the total budget is, $Q(x)=2.1x^2-3.5x+296$. The fraction of total budget spent on human resources is modeled by $\frac{P(x)}{Q(x)}$.
Therefore,
$\frac{P(x)}{Q(x)}=\frac{1.75x^2-15.9x+160}{2.1x^2-3.5x+296}$.
b. In $2010$, approximately $P(x)=2500$ and $Q(x)=3700$.
Therefore, by $\frac{2500}{3700}=\frac{25}{37}\times 100=67.567\%=68\%$
c. $2010$ is $40$ years after the year $1970$ which is at $x=0$.
Therefore, $x=40$
$\frac{P(40)}{Q(40)}=\frac{$1.75(40)^2-15.9(40)+160}{2.1(40)^2-3.5(40)+296}=\frac{2324}{3516}=66.1\%=66\%$.
As $66\%<68\%$, this result underestimates the actual percent found in part b) by $68\%-66\%=2\%$.
d. The rules of the horizontal asymptotes are as follows,
1. If the numerator's degree is less than the denominator's degree, there is a horizontal asymptote at $y = 0$.
2. If the numerator's degree equals the denominator's degree, there is a horizontal asymptote at $y = c$, where $c$ is the ratio of the leading terms or their coefficients.
3. If the numerator's degree is more than the denominator's degree, then there is no horizontal asymptote.
Since the degree of $P(x)$ equal the degree of $Q(x)$.
$y=\frac{1.75}{2.1}=0.83$
In time, the budget spent of human resources will approach $83\%$,
