Chapter 1, Problem 1E

To determine

The percentage of the atmosphere above each height in Table 1-1 using the rule.

Answer to Problem 1E

Height (km)	% Above
22.4	6.25%
16.8	12.5%
11.2	25%
5.6	50%
Sea level	100%

Explanation of Solution

A simple rule can be used to determine the rate at which density decreases with height. The percentage of atmosphere at sea level is $100\%$, and it will be halved for every $5.6\text{km}$ from its previous sea level.

$\text{Percentage of atmosphere at sea level=100\%}$

The expression for calculating the percentage of atmosphere at various heights from the sea level or previous height from the sea level using the above rule is as follows:

$\text{Percentage}\text{of}\text{atmosphere}\text{at}\text{a}\text{height}\text{from}\text{sea}\text{level}=\frac{\text{Percentage}\text{of}\text{atmosphere}\text{at}\text{sea}\text{level}}{2}$

Substitute percentage of atmosphere at sea level as $100\%$ to find the percentage of atmosphere at height of $5.6\text{km}$ from the sea level.

$\begin{array}{c}\text{Percentage}\text{of}\text{atmosphere}\text{at}\text{a}\text{height}\text{of}5.6\text{km}\text{from}\text{sea}\text{level}=\frac{\text{100\%}}{2}\\ =50\%\end{array}$

To calculate the percentage of atmosphere at the remaining heights from the sea level, the percentage of atmosphere at the previous level is used, and it can be given as follows:

$\text{\% of}\text{atmosphere}\text{at}\text{a}\text{height}\text{from}\text{sea}\text{level}=\frac{\%\text{of}\text{atmosphere}\text{at previous height from}\text{sea}\text{level}}{2}$

Substitute the given values of height to determine the percentage of atmosphere at that level. Then,

$\begin{array}{c}\text{Percentage}\text{of}\text{atmosphere}\text{at}\text{a}\text{height}\text{of}11.2\text{km}\text{from}\text{sea}\text{level}=\frac{\text{50\%}}{2}\\ =25\%\end{array}$

$\begin{array}{c}\text{Percentage}\text{of}\text{atmosphere}\text{at}\text{a}\text{height}\text{of}16.8\text{km}\text{from}\text{sea}\text{level}=\frac{\text{25\%}}{2}\\ =12.5\%\end{array}$

$\begin{array}{c}\text{Percentage}\text{of}\text{atmosphere}\text{at}\text{a}\text{height}\text{of}22.4\text{km}\text{from}\text{sea}\text{level}=\frac{\text{12}\text{.5\%}}{2}\\ =6.25\%\end{array}$