Problem $25.$ In a constant$-$temperature $T_0$ atmosphere, whose composition matches the Earth's,

as we derived in the class, the atmospheric pressure as a function of elevation is given by

$P(z)=P_0$exp$(\frac{-gz}{R_{\text{a}\text{i}\text{r}}{T}_0})$

where exp$(x)=e^x,P(z)$ is the pressure as a function of height, $z$ is the elevation, $P_0$ is the

pressure at $z=0,g$ is the gravitational acceleration, $R_{\text{a}\text{i}\text{r}}=0\mathrm{.}287k\frac{J}{kg*K},$ and $T_0$ is the

absolute temperature. If $P_0=101$kPa and $T_0=15C,$ calculate the pressure at $z=8,000m.$

What is the relative percent error between the pressure predicted by the simplified model and

the empirical data for the standard atmosphere from the table below? The relative percent

error is defined to be:

$(\frac{P_{\text{m}\text{o}\text{d}\text{e}\text{l}}-P_{\text{d}\text{a}\text{t}\text{a}}}{P_{\text{d}\text{a}\text{t}\text{a}}})100\%$

Table $1$: Properties of the Standard Atmosphere

$($a$)3\mathrm{.}2\%$

$($c$)9\mathrm{.}7\%$

$($b$)6\mathrm{.}5\%$

$($e$)$ None of these.

$($d$)14\mathrm{.}7\%$