Question

Atmospheric pressure decreases by about 11.8% for every 1000 meters you climb. The pressure at sea level is 1013 millibars (a unit of pressure). Construct an exponential model to represent the atmospheric pressure at a given altitude in thousands of meters. Use your model to determine the atmospheric pressure at an altitude of 4000 meters.
(Express your answer in millibars rounded correctly to the nearest tenth.)

Answer 1

at sea level, the pressure is 1013, that's when the altitude is at 0, sea level, let's see

$\bf \textit{Periodic Exponential Decay}\\\\ A=I(1 - r)^{\frac{t}{p}}\qquad \begin{cases} A=\textit{accumulated amount}\to &1013\\ I=\textit{initial amount}\\ r=rate\to 11.8\%\to \frac{11.8}{100}\to &0.118\\ t=\textit{meters climbed}\to &0\\ p=period\to &1000 \end{cases} \\\\\\ 1013=I(1-0.118)^{\frac{0}{1000}}\implies 1013=I\cdot 1\implies 1013=I$

so, the inital amount is 1013, when t = 0,

$\bf \textit{Periodic Exponential Decay}\\\\ A=I(1- r)^{\frac{t}{p}}\qquad \begin{cases} A=\textit{accumulated amount}\\ I=\textit{initial amount}\to &1013\\ r=rate\to 11.8\%\to \frac{11.8}{100}\to &0.118\\ t=\textit{meters climbed}\to &t\\ p=period\to &1000 \end{cases} \\\\\\ A=1013(1-0.118)^{\frac{t}{1000}}\implies A=1013(0.882)^{\frac{t}{1000}}$

now, to check the atmospheric pressure at 4000, simply set t = 4000, to get A.