Thread: Show that the qualities are related by the law?

1. Show that the qualities are related by the law?

I am studying analytical methods as part of my course and i have just enjoyed Pascals triangle and got my head round binomial theorem :-)
Can any one help me with the question below a bit lost!!

Atmospheric pressure(p) is measured at varying altitudes (h) and the results are
shown below
(h) meters (p) cm
500 73.39
1500 68.42
3000 61.6
5000 53.56
8000 43.41

Show that the quantities are related by the law
p=ae
kh

Where a and k are constants.
Determine the values of a and k and state the law.
Find the atmospheric pressure at 10,000m
many many thanks folks!! ;-)

I am studying analytical methods as part of my course and i have just enjoyed Pascals triangle and got my head round binomial theorem :-)
Can any one help me with the question below a bit lost!!

Atmospheric pressure(p) is measured at varying altitudes (h) and the results are
shown below
(h) meters (p) cm
500 73.39
1500 68.42
3000 61.6
5000 53.56
8000 43.41

Show that the quantities are related by the law
p=ae
kh

Where a and k are constants.
Determine the values of a and k and state the law.
Find the atmospheric pressure at 10,000m
many many thanks folks!! ;-)
This doesn't have anything to do with "Pascal's Triangle" or "Binomial Theorem". Perhaps that is what was throwing you off!

You want to show that the data fit $\displaystyle P= ae^{kh}$
Okay, if that is true then for the first set, P= 73.39 and h= 500 so $\displaystyle 73.39= ae^{500k}$ and for the last, P= 43.41 and h= 8000 so 43.41= ae^{8000k}. That gives two equations for a and k. We can eliminate a by dividing one equation by the other. $\displaystyle \frac{73.39}{43.41}= \frac{e^{500k}}{e^{8000k}}= e^{(500-8000)k}= e^{-7500k}$ or $\displaystyle e^{-7500k}= 1.6906$. Solve that by taking the natural logarithm of both sides, then put back into your first equation.

After you have found k, put it back into either of the original equations to solve for a. Then check the other values on your list to see if the satisfy that equation, at least approximately.

I chose the two ends of the list for P and h to find a and k because they are the extremes and tend to give a better approximation to the whole list than numbers close together.