Chapter 5.4, Problem 119AYU

Atmospheric Pressure 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 Junction

$p\left(h\right)=760e^{-0.145h}$

Find the height of an aircraft if the atmospheric pressure is $320$ millimeters of mercury.

Find the height of a mountain if the atmospheric pressure is $667$ millimeters of mercury.

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 $p\left(h\right)=760e^{-0.145h}$

Explanation:

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 $5.9655\text{kilometers}$.

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 $p\left(h\right)=760e^{-0.145h}$

Explanation:

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 $0.9002\text{kilometers}$.