Short Answer 
The atmospheric pressure can be modeled using the exponential function f(x) = 14.7 (1/2)^(x/3.6), where 14.7 lb/in² is the initial pressure at sea level and the pressure halves for every 3.6 miles ascended. This function allows us to calculate atmospheric pressure at various altitudes.
Step 1: Understand the Exponential Function
The general formula for modeling atmospheric pressure is the exponential function: f(x) = ab^x, where a represents the initial value and b is the growth rate. In this scenario, at sea level (0 miles), the atmospheric pressure is 14.7 lb/in². This value serves as the initial pressure for our function.
Step 2: Determine the Growth Rate
As I climb above sea level, the pressure reduces by half for every 3.6 miles ascended. Therefore, the growth factor can be calculated as (1/2)^(1/3.6). This adjustment allows us to model how quickly pressure decreases with altitude. The function now takes the form where initial pressure is 14.7 and the growth factor reflects this decline.
Step 3: Formulate the Pressure Function
The atmospheric pressure function based on height becomes f(x) = 14.7 (1/2)^(x/3.6). In this equation, x indicates the number of miles above sea level, while f(x) represents the atmospheric pressure in lb/in². From this function, we can calculate pressure values, such as:
- At 0 miles: 14.7 lb/in²
- At 3.6 miles: 7.35 lb/in²
- At 7.2 miles: 3.675 lb/in²
- At 10.8 miles: 1.8375 lb/in²