# Thread: Diffy Q (Separable Equations)

1. ## Diffy Q (Separable Equations)

The barometric pressure p (in inches of mercury) at an altitude x miles above sea level satisfies the initial value problem dp/dx = (-0.2)p, p(0) = 29.92. (a) Calculate the barometric pressure at 10,000 ft and again at 30,000 ft. (b) Without prior conditioning, few people can survive when the pressure drops to less than 15 in. of mercury. How high is that?

2. Originally Posted by bearej50
The barometric pressure p (in inches of mercury) at an altitude x miles above sea level satisfies the initial value problem dp/dx = (-0.2)p, p(0) = 29.92. (a) Calculate the barometric pressure at 10,000 ft and again at 30,000 ft. (b) Without prior conditioning, few people can survive when the pressure drops to less than 15 in. of mercury. How high is that?
You have $p' = (-0.2)p \implies \frac{p'}{p} = -0.2$.
Therefore, $[ \ln p ]' = -0.2 \text{ so }\ln p = -0.2x + k \implies p = Ae^{-0.2x}$.
But, $p(0) = 29.92 \implies Ae^{(-0.2)(0)} \implies A=29.92$.
Thus, $p(x) = 29.92 e^{-0.2x}$.

For the second problem you need to solve, $29.92 e^{-0.2x} = 15$.

3. thank you greatly