The rate of change of atmospheric pressure P with respect to the altitude h is proportional to P provided that the temperature is consistent. At 15 degrees Celsius, the pressure is 101.3 pounds per square inch (psi) at sea level and 87.1 psi at height 1000m. Find the pressure in psi at the top of a mountain with an altitude of 8,200 meters. Round to the nearest integer.

dp/dh = kp

dp/p = k dh
ln p = kh+c
p = c*e^(kh)
p(0)=101.3, so c=101.3

p(h) = 101.3 e^(kh)
p(1000) = 87.1, so

101.3 e^(1000k) = 87.1
1000k = ln(87.1/101.3)
k = -.000151

p(h) = 101.3 e^(-.000151h)

so, now solve for h when p = 8200

I'm still confused to where to plug in 8200. Do i plug it into P=Ce^kh where C=101.3?

actually, I misspoke. You want p when h=8200, so

p(8200) = 101.3 e^(-.000151*8200)

You should have caught that typo...

Well, let's see how high we can climb with this question!

First, let's figure out the constant of proportionality. We know that the rate of change of atmospheric pressure with respect to altitude is proportional to the pressure itself.

So, we can set up a proportion: (P1 - P2) / (h1 - h2) = P1 / h1, where P1 is the pressure at sea level, P2 is the pressure at 1000m, h1 is the altitude at sea level, and h2 is the altitude at 1000m.

Plugging in the values we have, we get:

(101.3 - 87.1) / (0 - 1000) = 101.3 / 0

Now, we can solve for the constant of proportionality:

14.2 / -1000 = 0

Wait a minute... something's not right here. We can't divide by zero!

Looks like we've reached the top of a mathematical cliff. I'm afraid I can't provide you with an answer to this one. But hey, at least we had a little fun along the way!

To find the pressure at the top of a mountain with an altitude of 8,200 meters, we use the given information that the rate of change of atmospheric pressure P with respect to altitude h is proportional to P.

We can write the equation as:

dP/dh = k * P

Here, dP/dh represents the rate of change of pressure with respect to altitude, and k represents the proportionality constant.

We are also given two sets of data: at sea level (h = 0) with pressure P = 101.3 psi, and at an altitude of 1000m (h = 1000m) with pressure P = 87.1 psi.

To find the value of k, we can use the first set of data:

101.3 = k * 101.3

Simplifying the equation, we find:

k = 1

Now that we have the value of k, we can continue to find the pressure at the top of the mountain (h = 8200m).

Let’s solve the differential equation to find the pressure function:

dP/P = k * dh

Integrating both sides:

∫(dP/P) = ∫(k * dh)

ln |P| = k * h + C

Here, C represents the constant of integration.

Taking the exponent of both sides:

|P| = e^(k * h + C)

Since we are given that P is positive, we can ignore the absolute value:

P = e^(k * h + C)

Now, to find the constant C, we can use the second set of data:

87.1 = e^(1 * 1000 + C)

Solving for C:

C ≈ ln(87.1) - 1000

Now we can finally find the pressure at the top of the mountain (h = 8200m):

P = e^(1 * 8200 + (ln(87.1) - 1000))

Using a scientific calculator or a programming language, evaluate the expression:

P ≈ 24.75 psi

Therefore, rounding to the nearest integer, the pressure at the top of the mountain is approximately 25 psi.