Calculus Help?

Question

Q: The pressure of the air in the Erth's atmosphere decreases exponentially with altitude above the surface of the earth. The pressure at the earths surface (sea level) is about 14.7 pounds per square inch and the pressure at 2000 is approximately 13.5psu.

Write a function expressing pressure in terms of altitude.

Please show me what you did.

Any help will be greatly appreciated. Please dont think that you are doing my homework. I have another 23 questions and have already done ten. I am just stuck on this one.

Thank you for your help

Mike

Please Help

Answer 1

P - pressure
h - altitude.
a and b - constants.

Since the pressure is going down as height goes up the function will be a negative exponential and one of the form
P = ae^(-kh).
Now find a and b. You do that by putting in the values at your two points.
At zero height the pressure is 14∙7, so e^-kh must equal 1.
→ P = 14.7*e^(-kh).
→a = 14∙7

Now with information for the other point, calculate b.
P = 14.7*e^(-kh)
13.5 = (14∙7)e^(-b*2000)
13.5/ 14∙7 = e^(-b*2000)
ln(13.5/ 14∙7) = -b*2000
-0∙085 157 7808 = -b*2000
b = -0∙085 157 7808 / -2000
b = 0∙0000 425 78


P = 14.7*e^(-0.0000425 78*h).

Answer 2

Since the pressure is decreasing with height the function will be a negative exponential one of the form P = ae^(-kh).
Now a must be 14.7 because when h = 0 then e^0 = 1.
So we have P = 14.7*e^(-kh).
You also know that when h = 2000 then P = 13.5 so
13.5 = 14.7*e^(-2000k)
Taking natural logarithms (written ln) of this line you get
ln(13.5) = ln(14.7) - 2000k
This is using the rules that
ln(pq) = ln(p) + ln(q) and ln(p^q) = q*ln(p) and that ln(e) = 1
The earlier line can be changed to 2000k = ln14.7 - ln13.5
You will have to take it from there as I don't have a scientific calculator with natural logarithms on it with me.

Answer 3

Exponential function decrease is:

p = a*e^(-b*h), where p - pressure, h - altitude, a and b - constants.
All you need is to find a and b. You do that by plugging values at your two points. Therefore:

14.7 = a, (h = 0) and
13.5 = a*e^(-b*2000).

From there b = ln(14.7/13.5)/2000 = 4.26*10^(-5), and your function is:

p = 14.7*e^(-0.0000426*h).

Answer 4

First of all, you aren't given enough information. Two points can draw only a linear relationship. You require more information, whether it be points or properties. Also, when dealing with a problem of this type, don't forget that you must use absolute pressure, not relative pressure.

Answer 5

p = a exp( -kh)
14.7 = a exp(-k0) = a
13.5 = a exp(-k2) = 14.7 exp(-k2)
so 2k = ln (14.7/13.5) = 0.4258
p = 14.7 exp( -0.4258h) = 14.7 / 1.0435^h
where p is in psi and h is in Km