Exponential decay

Question

The kilopascal is a unit of measure for atmospheric pressure Suppose the atmospheric air pressure at sea level is about 102 kilopascals. For every 1000-m increase in altitude, the pressure decreases about 11.6% What is the approximate pressure at an altitude of 4000 m?

Answer 1

We are told that every 1000m increase in altitude results in dropping the pressure by 11.6%. This means that the pressure at 1000m must be 88.4% of what it was at sea level. In other words, the pressure at 1000m is

P(1000) = 0.884(102 kPa) = 90.168 kPa.

If we travel another 1000m, the pressure at 2000m will again be 88.4% of what it was at 1000 so that

P(2000) = 0.884(90.168 kPa) = 79.708512 kPa.

If we travel another 1000m, the pressure at 3000m will again be 88.4% of what it was at 2000 so that

P(3000) = 0.884(79.708512 kPa) = 70.462324608 kPa.

If we travel another 1000m, the pressure at 4000m will again be 88.4% of what it was at 3000 so that

P(4000) = 0.884(70.462324608 kPa) = 62.2886949535 kPa.

Since we've reached our final answer, we can now round to 3 significant figures to get 62.3 kPa.

In general, the pressure at any altitude in this case will be P(z) = 102(0.884)^z/1000, where z is measured in meters and P is measured in kPa.