Question

the formula $p = 14.7e^{-0.21x}$ gives the average atmospheric pressure p, in pounds per square inch, at an altitude x, in miles above sea - level. find the elevation of a jet if the atmospheric pressure outside the jet is $7.9 lb/in^{2}$. the elevation of the jet is approximately miles. (do not round until the final answer. then round to the nearest tenth as needed.)

Explanation:

Step1: Substitute given values

Given $P = 7.9$ and $P=14.7e^{- 0.21x}$, we substitute $P$:
$7.9 = 14.7e^{-0.21x}$

Step2: Isolate the exponential term

Divide both sides by 14.7:
$\frac{7.9}{14.7}=e^{-0.21x}$
$e^{-0.21x}=\frac{7.9}{14.7}\approx0.5374$

Step3: Take natural - logarithm of both sides

$\ln(e^{-0.21x})=\ln(0.5374)$
Using the property $\ln(e^{a}) = a$, we get:
$-0.21x=\ln(0.5374)$

Step4: Solve for $x$

$x=\frac{\ln(0.5374)}{- 0.21}$
Since $\ln(0.5374)\approx - 0.621$, then $x=\frac{-0.621}{-0.21}\approx2.957$

Step5: Round to the nearest tenth

Rounding $2.957$ to the nearest tenth gives $3.0$

Answer:

$3.0$