Question

Devise an exponential decay function that fits the given​ data, then answer the accompanying questions. Be sure to identify the reference point ​(tequals ​0) and units of t. The pressure of a certain​ planet's atmosphere at sea level is approximately 800 millibars and decreases exponentially with elevation. At an elevation of 35 comma 000 ​ft, the pressure is​ one-third of the​ sea-level pressure. At what elevation is the pressure half of the​ sea-level pressure? At what elevation is it 2 ​% of the​ sea-level pressure?

Answer 1

22145.27733 ft

124984.76055 ft

Step-by-step explanation:

The equation of pressure is

`P=P_0e^(-kh)`

where,

`P_0` =Atmospheric pressure = 800 mbar

k = Constant

h = Altitude = 35000 ft

`P=(1)/(3)P_0`

`(1)/(3)P_0=P_0e^(-k35000)\\\Rightarrow (1)/(3)=e^(-k35000)\\\Rightarrow 3=e^(k35000)\\\Rightarrow ln3=k35000\\\Rightarrow k=(ln3)/(35000)\\\Rightarrow k=3.13* 10^(-5)`

Now

`P=(1)/(2)P_0`

`ln2=kh\\\Rightarrow h=(ln2)/(k)\\\Rightarrow h=(ln2)/(3.13* 10^(-5))\\\Rightarrow h=22145.27733\ ft`

The altitude will be 22145.27733 ft

`P=0.02P_0`

`0.02P_0=P_0e^(-kh)\\\Rightarrow 0.02=e^{-3.13* 10^(-5)h}\\\Rightarrow ln0.02=-3.13* 10^(-5)h\\\Rightarrow h=(ln0.02)/(-3.13* 10^(-5))\\\Rightarrow h=124984.76055\ ft`

The elevation is 124984.76055 ft