Solve the problem.
-The barometric pressure $\mathrm { p }$ at an altitude of $\mathrm { h }$ miles above sea level satisfies the differential equation $\frac { d p } { d h } = - 0.2 p$. If the pressure at sea level is $29.92$ inches of mercury, find the barometric pressure at $29,000 \mathrm { ft }$.
A) 4.99 in.
B) 9.97 in.
C) 89.75 in.
D) 0.09 in.

Question

Quizplus · Accepted Answer

First, we need to convert the altitude from feet to miles.

$29,000 	ext{ ft} = \dfrac{29,000}{5,280} 	ext{ miles} \approx 5.5 	ext{ miles}$

Now we can solve the differential equation:

$\dfrac{dp}{dh} = -0.2p \Rightarrow \dfrac{dp}{p} = -0.2dh$

Integrating both sides:

$\ln|p| = -0.2h + C$

We can find the value of the constant $C$ by using the initial condition that the pressure at sea level is 29.92 inches of mercury:

$\ln|29.92| = -0.2(0) + C \Rightarrow C = \ln|29.92|$

So the equation for the barometric pressure as a function of altitude is:

$\ln|p| = -0.2h + \ln|29.92| = \ln\left(\dfrac{29.92}{e^{0.2h}}ight)$

Now we can use this equation to find the pressure at an altitude of 5.5 miles.

$p = \pm e^{\ln\left(\dfrac{29.92}{e^{0.2(5.5)}}ight)} \approx 9.97 	ext{ in}$

Since pressure cannot be negative, we take the positive solution of the equation, which gives us a pressure of approximately 9.97 inches of mercury at an altitude of 29,000 feet. Therefore, the answer is B.

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