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.