Answer
a) See below.
b) The linear function that models the life expectancy of women is $y=0.17x+73$.
c) The life expectancy of American women born in 2020 is 83.2 years.
Work Step by Step
(b)
If $x$ denotes the number of years after 1960, then the year 1970 denotes 10 and 2000 denotes 40.
From the graph, for the year 1970, the point is $\left( {{x}_{1}},{{y}_{1}} \right)=\left( 10,74.7 \right)$ and, for the year 2000 the point is $\left( {{x}_{2}},{{y}_{2}} \right)=\left( 40,79.7 \right)$.
Now, substitute $\left( {{x}_{1}},{{y}_{1}} \right)=\left( 10,74.7 \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)=\left( 40,79.7 \right)$ in the line equation:
$\begin{align}
& y-74.7=\frac{79.7-74.7}{40-10}\left( x-10 \right) \\
& y-74.7=\frac{5}{30}\left( x-10 \right) \\
& y-74.7=0.17\left( x-10 \right) \\
& y-74.7=0.17x-1.7
\end{align}$
Simplify further,
$\begin{align}
& y=0.17x+74.7-1.7 \\
& y=0.17x+73
\end{align}$
Hence, the linear function that models the life expectancy of women is $y=0.17x+73$.
(c)
Substitute the value of $x=2020-1960=60$ in the equation obtained in part (b).
That is,
$\begin{align}
& y=0.17\times 60+73 \\
& =10.2+73 \\
& =83.2
\end{align}$
Hence, the life expectancy of American women born in 2020 is 83.2 years.
