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 $6.8 lb/in.^{2}$. the elevation of the jet is approximately $square$ miles. (do not round until the final answer. then round to the nearest tenth as needed.)

Explanation:

Step1: Substitute given pressure value

We are given $P = 6.8$ and the formula $P = 14.7e^{- 0.21x}$. So, $6.8=14.7e^{-0.21x}$.

Step2: Isolate the exponential term

Divide both sides by 14.7: $\frac{6.8}{14.7}=e^{-0.21x}$, which simplifies to $\frac{68}{147}=e^{-0.21x}$.

Step3: Take natural - logarithm of both sides

$\ln(\frac{68}{147})=\ln(e^{-0.21x})$. Since $\ln(e^{a}) = a$, we have $\ln(\frac{68}{147})=-0.21x$.

Step4: Solve for x

$x =-\frac{\ln(\frac{68}{147})}{0.21}$. We know that $\ln(\frac{68}{147})=\ln(68)-\ln(147)$. $\ln(68)\approx4.2207$ and $\ln(147)\approx4.9905$. So, $\ln(\frac{68}{147})\approx4.2207 - 4.9905=-0.7698$. Then $x=\frac{0.7698}{0.21}\approx3.7$.

Answer:

$3.7$