Answer
(a)
The median length of life for people who reach 65, that
is, the age for which the survival rate is 0.50. is $87$ years.
(b)
Find the age beyond which virtually nobody lives. (There
are, of course, exceptions.) is $98$ years
Work Step by Step
the survival function for life after 65 is approximately given by:
$$
S(x)=1-0.058 x-0.076 x^{2}
$$
where $x$ is measured in decades. This function gives the probability that an individual who reaches the age of 65 will live at
least $x$ decades ($10x $ years) longer.
(a). Find the median length of life for people who reach 65, that
is, the age for which the survival rate is 0.50.
$$
\begin{aligned}
0.50 &=1-0.058 x-0.076 x^{2}\\
\Rightarrow\\
0.076 x^{2}+0.058 x -1+ 0.50 &=0\\
0.076 x^{2}+0.058 x - 0.50 &=0\\
\Rightarrow\\
76 x^{2}+58 x - 500 &=0\\
38x^{2}+29x - 250&=0\\
\end{aligned}
$$
This does not appear to factor, so we’ll try the quadratic formula:
$$
\begin{aligned}
x&=\frac{-29 \pm \sqrt{(29)^{2}-4(38)(-250)}}{2(38)}\\
&=\frac{-29 \pm \sqrt{38,841}}{76}\\
x&= \frac{-29-\sqrt{38,841}}{76} \approx-2.97 \\
& \text{and}\\
x&=\frac{-29+\sqrt{38,841}}{76} \approx 2.21\\
& \quad\quad\left[\begin{array}{c}{ \text {We ignore the negative value }} \\ {\text { then} }\end{array}\right] , \\
x&=2.2
\end{aligned}
$$
decades $10. x= 10. (2.2)=22 $ years, so
The median length of life is 65+22=87 years.
(b) Find the age beyond which virtually nobody lives. (There
are, of course, exceptions.)
If nobody lives, $S(x)=0$
$$
\begin{aligned}
0 &=1-0.058 x-0.076 x^{2}\\
\Rightarrow\\
0.076 x^{2}+0.058 x -1 &=0\\
76 x^{2}+58 x - 1000 &=0\\
38x^{2}+29x - 500&=0\\
\end{aligned}
$$
This does not appear to factor, so we’ll try the quadratic formula:
$$
\begin{aligned}
x &=\frac{-29 \pm \sqrt{(29)^{2}-4(38)(-500)}}{2(38)}\\
&=\frac{-29 \pm \sqrt{76,841}}{76}\\
x&\frac{-29-\sqrt{76,841}}{76} \approx-4.03 \\
& \text{and}\\
x&=\quad \frac{-29+\sqrt{76,841}}{76} \approx 3.27\\
& \quad\quad\left[\begin{array}{c}{ \text {We ignore the negative value }} \\ {\text { then} }\end{array}\right] , \\
x&=3.3
\end{aligned}
$$
decades $10. x= 10. (3.3)=33 $ years, so
The median length of life is 65+33=98 years.
