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The atmospheric pressure P, in pounds per square inch (psi), is given by P = 14.7e^{-0.21a},...

Question:

The atmospheric pressure, {eq}P {/eq}, in pounds per square inch (psi), is given by

{eq}P = 14.7e^{-0.21a} {/eq}

where {eq}a {/eq} is the altitude above sea level (in miles).

If a city has an atmospheric pressure of {eq}12.31 {/eq} psi, what is its altitude? (Recall that {eq}1 \textrm{ mi} = 5\textrm{,}280 \textrm{ ft} {/eq}. Round your answer to the nearest foot.)

Using Logarithms

To solve the given question, we need to makes use of logarithms. As the base is {eq}e {/eq}, we can make use of the natural log as it will be easier to work with. This is shown below.

Answer and Explanation:


We can find the value of a when P=12.31 as follows.

$$\begin{align} &P = 14.7e^{-0.21a}=12.31\\ &e^{-0.21a}=0.8374\\ &\text{Taking the natural of both the sides,}\\ &-0.21a=\ln (0.8374)&&&&\left [ \because \ln (e^x)=x\right ]\\ \therefore &a=0.845\,\text{mi} \end{align} $$


The altitude in feet will be:

$$\begin{align} 0.845*5280\approx 4462\,\text{feet}\\ \end{align} $$


Learn more about this topic:

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What is a Logarithm?

from Math 101: College Algebra

Chapter 10 / Lesson 3
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