Chapter 7.4, Problem 17E

a.

To determine

To calculate : the pressure at an altitude of $3000m$

Answer to Problem 17E

The pressure at an altitude of $3000m$ is $64.59kPa$

Explanation of Solution

Given information : The question is

The rate of change of atmospheric pressure $P$ with respect to altitude $h$ is proportional to $P$ , provided that the temperature is constant. At ${15}^{\circ }C$ the pressure is $101.3kPa$ at sea level and $87.14kPa$ at $h=1000m$

Calculation : the change in $P$ with respect to $h$ is proportional to $P$

  $\begin{array}{l}\frac{dP}{dh}=kP\\ P\left(h\right)=P\left(0\right)e^{kh}\\ P\left(1000\right)=P\left(0\right)e^{k1000}\\ 87.14=101.3e^{k1000}\\ e^{k1000}=\frac{87.14}{101.3}=0.86\\ \ln \left(e^{k1000}\right)=\ln \left(0.86\right)\\ k\cdot 1000=\ln \left(0.86\right)\\ k=\frac{\ln \left(0.86\right)}{1000}=-0.00015\\ P\left(h\right)=P\left(0\right)e^{kh}\\ P\left(3000\right)=101.3e^{-0.00015\cdot 3000}\\ P\left(3000\right)=101.3e^{-0.45}\\ P\left(3000\right)=64.59kPa\end{array}$

b.

To determine

To calculate : the pressure at the top of mount McKinley at an altitude of $6187m$

Answer to Problem 17E

The pressure at the top of mount McKinley at an altitude of $6187m$ is $40.046$

Explanation of Solution

Given information : The question is

The rate of change of atmospheric pressure $P$ with respect to altitude $h$ is proportional to $P$ , provided that the temperature is constant. At ${15}^{\circ }C$ the pressure is $101.3kPa$ at sea level and $87.14kPa$ at $h=1000m$

Calculation :

  $\begin{array}{l}P\left(h\right)=P\left(0\right)e^{kh}\\ P\left(6187\right)=101.3e^{-0.00015\cdot 6187}\\ P\left(6187\right)=40.046m\end{array}$