Chapter 5.4, Problem 119AYU

(a)

To determine

The height of an aircraft if the atmospheric pressure is $320$ millimeters of the mercury, where the pressure measured in millimeters of mercury is related to the height $h$ (in kilometers) above sea level by the function $p\left(h\right)=760e^{-0.145h}$

Answer to Problem 119AYU

Solution:

The height of an aircraft, if the atmospheric pressure is $320$ millimeters of the mercury is $5.9655\text{kilometers}$ .

Explanation of Solution

Given information:

The atmospheric pressure $p$ on an object decreases with increasing height.

This pressure, measured in millimeters of mercury is related to the height $h$ (in kilometers) above sea level by the function $p\left(h\right)=760e^{-0.145h}$

From the given information,

The pressure measured in millimeters of mercury is given by the function

  $p\left(h\right)=760e^{-0.145h}$

Here, the atmospheric pressure is $320$ millimeters of the mercury

Thus, $p=320$

  $\Rightarrow 320=760e^{-0.145h}$

Divide both sides by $760$ ,

  $\Rightarrow \frac{320}{760}=e^{-0.145h}$

  $\Rightarrow \frac{8}{19}=e^{-0.145h}$

Taking natural log on both sides,

  $\Rightarrow \ln \left(\frac{8}{19}\right)=\ln \left(e^{-0.145h}\right)$

  $\Rightarrow \ln \left(\frac{8}{19}\right)=-0.145h$

  $\Rightarrow h=5.9655\text{kilometers}$

Thus, the height of an aircraft if the atmospheric pressure is $320$ millimeters of the mercury is $5.9655\text{kilometers}$ .

(b)

To determine

The height of a mountain, if the atmospheric pressure is $667$ millimeters of the mercury, where the pressure measured in millimeters of mercury is related to the height $h$ (in kilometers) above sea level by the function $p\left(h\right)=760e^{-0.145h}$

Answer to Problem 119AYU

Solution:

The height of a mountain, if the atmospheric pressure is $667$ millimeters of the mercuryis $0.9002\text{kilometers}$

Explanation of Solution

Given information:

The atmospheric pressure $p$ on an object decreases with increasing height.

This pressure, measured in millimeters of mercury is related to the height $h$ (in kilometers) above sea level by the function $p\left(h\right)=760e^{-0.145h}$

From the given information,

The pressure measured in millimeters of mercury is given by the function,

  $p\left(h\right)=760e^{-0.145h}$

Here, the atmospheric pressure is $667$ millimeters of the mercury

Thus, $p=667$

  $\Rightarrow 667=760e^{-0.145h}$

Divide both sides by $760$ ,

  $\Rightarrow \frac{667}{760}=e^{-0.145h}$

Taking natural log on both sides,

  $\Rightarrow \ln \left(\frac{667}{760}\right)=\ln \left(e^{-0.145h}\right)$

  $\Rightarrow h=0.9002\text{kilometers}$

Thus, the height of a mountain, if the atmospheric pressure is $667$ millimeters of the mercury is $0.9002\text{kilometers}$ .