Chapter 4.3, Problem 48PPS

To determine

The formula (relation) between the barometric pressure and altitude.

Answer to Problem 48PPS

The formula (relation) is $y=-70x+724$ . Here $\left(y\right)$ is barometric pressure and $\left(x\right)$ is altitude.

Explanation of Solution

Given:It is given that the barometric pressure is a linear function of altitude, which means the relationship between barometric pressure and altitude will be linear. Also, the two reading are observed as $\left(1.8,598\right)$ and $\left(2.1,577\right)$ where x-coordinate represents the altitude and y-coordinate represents the barometric pressure.

Calculation:

From the observation of two points $\left(1.8,598\right)$ and $\left(2.1,577\right)$ the slope $\left(m\right)$ is evaluated as,

  $\begin{array}{c}\left(m\right)=\frac{577-598}{2.1-1.8}\\ =-70\end{array}$

Now, considering that $\left(x\right)$ represents the altitude and $\left(y\right)$ represents the barometric pressure. Then, the relationship between barometric pressure $\left(y\right)$ and altitude $\left(x\right)$ is obtained as,

  $\begin{array}{c}y-y_1=m\left(x-x_1\right)\mathrm{......}\left(\text{Point-slope form}\right)\\ y-598=-70\left(x-1.8\right)\mathrm{......}\left(\begin{array}{l}\text{Here }{m}_{GH}=-70\\ \left(x_1,y_1\right)\to \left(1.8,598\right)\end{array}\right)\\ y-598=-70x+126\\ \text{or, }y=-70x+\mathrm{724......}\left(1\right)\end{array}$

To determine

The altitude if pressure is 657 millimetre of mercury.

Answer to Problem 48PPS

The altitude $\left(x\right)$ is equal to 0.957km.

Explanation of Solution

Given: It is given that the barometric pressure is a linear function of altitude, which means the relationship between barometric pressure and altitude will be linear. Also, the two reading are observed as $\left(1.8,598\right)$ and $\left(2.1,577\right)$ where x-coordinate represents the altitude and y-coordinate represents the barometric pressure.

Calculation:

From (1), the relationship between barometric pressure $\left(y\right)$ and altitude $\left(x\right)$ is obtained as $y=-70x+724$ . Therefore, when barometric pressure $\left(y\right)$ is 657 millimetre of mercury then the altitude $\left(x\right)$ from $y=-70x+724$ can be evaluated as,

  $\begin{array}{c}y=-70x+\mathrm{724......}\left(\text{From }\left(1\right)\right)\\ \left(657\right)=-70x+\mathrm{724......}\left(y=657\right)\\ \text{or, }70x=67\\ x=\mathrm{0.957......}\left(\text{Approx}\text{.}\right)\end{array}$

Hence, the altitude $\left(x\right)$ is equal to 0.957km.