The model is P(x) = 29.9(0.89)^x, where x is altitude in kilometers and P(x) is pressure in inches of mercury. Interpret this step by step:
- Starting value (sea level): P(0) = 29.9(0.89)^0 = 29.9 inHg. This is the pressure at 0 km.
- Multiplier per km: For each increase of 1 km in altitude, pressure is multiplied by 0.89. That means the pressure at x+1 km is 0.89 times the pressure at x km.
- Percent change per km: Multiplying by 0.89 is the same as a decrease of 1 - 0.89 = 0.11, i.e. an 11% decrease in pressure for each additional kilometer of altitude.
Examples:
- P(1) = 29.9(0.89) = 26.611 inHg (a drop of about 3.29 inHg, which is ~11%).
- P(5) = 29.9(0.89)^5 ≈ 29.9(0.5584) ≈ 16.70 inHg.
Optional: the instantaneous rate of change (derivative) is dP/dx = 29.9(0.89)^x ln(0.89). Since ln(0.89) ≈ -0.1165, the instantaneous relative rate is about -11.65% per km, which is the continuous-rate interpretation of the exponential model.
Summary: According to the model, atmospheric pressure decreases exponentially with altitude: it starts at 29.9 inHg at sea level and falls to about 89% of its previous value for every additional kilometer (about an 11% decrease per km).