MHB Interpreting Constant A & Deriving k in Air Pressure Decay

[fuzz95]

hey guys! I'm really confused as to what this question is trying to ask me!
can someone help me out :)

Air pressure decays approximately exponentially at about 0.4 per cent for each rise of 30 metres above sea level. If we let p = p(h) denote air pressure (measured in some appropriate units) at h metres above sea level, then we can model this phenomenon using the equation: p = p(h) = Aekh for some appropriate constants A and k.

Give an interpretation for the constant A. (We never need to know the actual numerical value of A to do the rest of this exercise.)
?and how do i derive this k = ln(0.996) / 30 ??thanks!

 

[Evgeny.Makarov]

Hi, and welcome to the forum!

If we let p = p(h) denote air pressure (measured in some appropriate units) at h metres above sea level, then we can model this phenomenon using the equation: p = p(h) = Aekh for some appropriate constants A and k.

This must be $p(h)=Ae^{kh}$. In plain text, it is customary to write exponentiation using ^, and don't forget the order of operations: exponentiation is done before multiplication. Therefore, in plain text p(h) can be written as A*e^(kh).

Give an interpretation for the constant A.

Hint: At what height the pressure is $A$?

and how do i derive this k = ln(0.996) / 30 ??

The statement "Air pressure decays approximately exponentially at about 0.4 per cent for each rise of 30 metres above sea level" amounts to
\[
p(h+30)=0.996p(h).
\]
Substitute the definition of $p$ and see if you can solve this equation for $k$.

 

[fuzz95]

okay so i was able to do the first part!

however still have no idea how to go about doing the second half??!

 

[Evgeny.Makarov]

however still have no idea how to go about doing the second half??!

The statement "Air pressure decays approximately exponentially at about 0.4 per cent for each rise of 30 metres above sea level" amounts to
\[
p(h+30)=0.996p(h).
\]
Substitute the definition of $p$ and see if you can solve this equation for $k$.

Well, let me do this for you. Substituting $p(h)=Ae^{kh}$ into the equation above gives
\[
Ae^{k(h+30)}=0.996Ae^{kh}.
\]
Can you solve it for $k$? If not, do you know basic properties of exponeniation? Do you know that $\ln(x)$ is the inverse of $e^x$? Have you seen any exponential equations solved? Have you solved any yourself? You may need to do some preparatory work.