The barometric pressure $\mathrm { p }$ at an altitude of $\mathrm { h }$ miles above sea level satisfies the differential equation $\frac { \mathrm { dp } } { \mathrm { dh } } = - 0.2 \mathrm { p }$. If the pressure at sea level is $29.92$ inches of mercury, find the barometric pressure at $18,000 \mathrm { ft }$.
A) 7.57 in.
B) 0.82 in.
C) 59.17 in.
D) 15.13 in.

Question

Quizplus · Accepted Answer

The given differential equation is separable and can be solved by integrating both sides. The solution to the differential equation is $p = p_0 e^{-0.2h}$, where $p_0 = 29.92$ inches of mercury is the pressure at sea level. To find the pressure at 18,000 ft, convert feet to miles (since the altitude $h$ is given in miles) by dividing by 5,280 (since there are 5,280 feet in a mile): $18,000 \, \text{ft} = \frac{18,000}{5,280} \, \text{miles} = 3.41 \, \text{miles}$. Substituting $h = 3.41$ and $p_0 = 29.92$ into the solution gives $p = 29.92 e^{-0.2 \times 3.41}$, which calculates to approximately $15.13$ inches of mercury.

The Barometric Pressure p\mathrm { p }p At an Altitude Of h\mathrm { h }h

The Barometric Pressure $\mathrm { p }$ At an Altitude Of $\mathrm { h }$