2- The earth’s atmospheric pressure p is often modeled by assuming that the rate dpdhat Which p changes with the altitude above sea level is proportional to hp. Suppose that the pressure at sea level is 1013 millibars (about 14.7 pounds per square inch) and that the pressure at an altitude of km is millibars). 2050a) Solve the equation dpkpdh=( = constant) to express kp in terms of . Determine the values of and the constant of integration from the given initial conditions. hkb) What is the atmospheric pressure at 50h =km? c) At what altitude is the pressure equal to 900 millibars? d) At what altitude is the pressure equal to half it’s value at the sea level?

You need to supply the missing numbers in "the pressure at an altitude of ___km is ___ millibars". Also, the stated assumption that the rate of change dp/dh is proportional to h p is incorrect.

I apologize for the incorrect assumption mentioned in the question. However, I can still explain how to approach this type of problem.

To solve the equation dp/dh = k * p, we can use separation of variables. Let's denote the constant of proportionality as k. We have:

dp/p = k * dh

Integrating both sides with respect to their respective variables, we get:

ln(p) = k * h + C

Where C is the constant of integration. Now, if we exponentiate both sides:

p = e^(k * h + C)

Using the rule of exponents, we can rewrite this as:

p = e^(k * h) * e^C

Introducing another constant A, such that A = e^C, we have:

p = A * e^(k * h)

Now, let's proceed to solve the given questions.

a) To determine the value of k, we need to use the initial condition provided. If the pressure at sea level (h = 0) is 1013 millibars, then our equation becomes:

1013 = A * e^(k * 0)
1013 = A * e^0
1013 = A

Therefore, A = 1013. Now our equation is:

p = 1013 * e^(k * h)

b) To find the atmospheric pressure at h = 50 km, we can substitute the given value into our equation:

p = 1013 * e^(k * 50)

c) To find the altitude at which the pressure is equal to 900 millibars, we can set p = 900 and solve for h:

900 = 1013 * e^(k * h)

d) To find the altitude at which the pressure is half its value at sea level, we can set p = 1013/2 and solve for h:

1013/2 = 1013 * e^(k * h)

To solve these equations and determine the values of k, h, and the constant of integration C in the context of your specific problem, you will need to provide the missing values for the pressure at a given altitude.