What would be the height of the atmosphere if the air density (a) were uniform and (b) decreased linearly to zero with height? Assume that at sea level the air pressure is 1.0 atm and the air density is 1.3 kg/`m^(3)`

Explanation

To find the height of the atmosphere under two different conditions of air density, we can break the problem down into two parts.

### Part (a): Uniform Air Density

1. **Identify Given Values**:
- Air pressure at sea level, \( P_0 = 1 \, \text{atm} = 1.01 \times 10^5 \, \text{Pa} \)
- Air density at sea level, \( \rho = 1.3 \, \text{kg/m}^3 \)
- Acceleration due to gravity, \( g = 9.81 \, \text{m/s}^2 \)
...