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Chapter 9: Problem 41

Earth's atmospheric pressure decreases by a factor of onehalf for every \(5.5-\mathrm{km}\) increase in altitude above sea level. Construct a plot of pressure versus altitude, assuming the pressure at sea level is one atmosphere (1 atm). Discuss the characteristics of your graph. At what altitude is the atmospheric pressure equal to \(0.001\) atm?

Short Answer

Expert verified
The graph of pressure vs altitude is a decreasing exponential curve. The atmospheric pressure is \(0.001\) atm at approximately \(48.17\) kilometers above sea level.

Step by step solution

01

Understanding The Given Information

From the given problem, it is understood that the pressure decreases by a factor of one-half for every \(5.5-\mathrm{km}\) increase in altitude above sea level. In other words, every \(5.5-\mathrm{km}\) upward, the atmospheric pressure is \(0.5 \times\) the pressure it was previously.
02

Defining The Atmospheric Pressure Function

The pressure can be represented as \(P = P_0 * (0.5)^{h/5.5}\). Where \(P\) represents the atmospheric pressure at a certain height \(h\), \(P_0\) is the atmospheric pressure at sea level which is 1 atm and the power function \(0.5^{h/5.5}\) is the factor representing the rate of decrease of the pressure as the altitude changes.
03

Constructing The Pressure vs Altitude Graph

To plot the graph of the function, take the altitude \(h\) on the x-axis and pressure \(P\) on the y-axis. Make sure to plot the exponential function as discussed previously. The plot should show a rapidly decreasing pressure as we raise the altitude.
04

Discussing The Graph Characteristics

Notice that the graph of the function is a decreasing exponential graph, starting from 1 atm at sea level and tending towards zero as the altitude increases. The rate of decrease is rapid at first and then starts to slow down as we ascend higher above sea level. This clearly represents the fact that the Earth's atmosphere thins out the higher you ascend.
05

Calculating The Altitude For A Specific Pressure

To find the altitude at which the atmospheric pressure is \(0.001\) atm, we substitute \(P = 0.001\) in the equation and solve for \(h\). Thus, \(0.001 = 1 * (0.5)^{h/5.5}\). Taking the logarithm of both sides, we get \(h = 5.5 \times \log_{0.5}^{0.001}\). After calculating, we find the altitude to be approximately \(48.17 \, km\) above sea level.

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