Question

atmospheric pressure $p$ in pounds per square inch is represented by the formula $p = 14.7e^{-0.21x}$, where $x$ is the number of miles above sea level. to the nearest foot, how high is the peak of a mountain with an atmospheric pressure of 8.305 pounds per square inch? hint: there are 5280 feet in a mile.

Explanation:

Step1: Substitute given pressure into formula

Given $P = 8.305$ and $P = 14.7e^{-0.21x}$, we get $8.305=14.7e^{-0.21x}$.

Step2: Solve for $e^{-0.21x}$

Divide both sides by 14.7: $\frac{8.305}{14.7}=e^{-0.21x}$, so $e^{-0.21x}\approx0.565$.

Step3: Take natural - logarithm of both sides

$\ln(e^{-0.21x})=\ln(0.565)$. Since $\ln(e^{a}) = a$, we have $-0.21x=\ln(0.565)$.

Step4: Solve for $x$ (in miles)

$x=\frac{\ln(0.565)}{- 0.21}$. Calculate $\ln(0.565)\approx - 0.57$, then $x=\frac{-0.57}{-0.21}\approx2.714$ miles.

Step5: Convert miles to feet

Since 1 mile = 5280 feet, the height in feet is $h = 2.714\times5280\approx14330$ feet.

Answer:

14330 feet