a.

Let «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mi»h«/mi»«/mfenced»«/mrow»«/mstyle»«/math» be the temperature at sea level, in degrees Celsius. Since the sea level temperature is independent of altitude, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mi»h«/mi»«/mfenced»«mo»=«/mo»«mn»20«/mn»«/mrow»«/mstyle»«/math».

b.

Since the drop in temperature is directly proportional to altitude, it can be represented in the form «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»g«/mi»«mfenced»«mi»h«/mi»«/mfenced»«mo»=«/mo»«mi»k«/mi»«mi»h«/mi»«/mrow»«/mstyle»«/math», where «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»g«/mi»«mfenced»«mi»h«/mi»«/mfenced»«/mrow»«/mstyle»«/math» is the change in air temperature, in degrees Celsius, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»k«/mi»«/mstyle»«/math» is the rate of change of the air temperature, and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»h«/mi»«/mstyle»«/math» is the altitude, in metres. Since the rate of change is known, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»0«/mn»«mo».«/mo»«mn»6«/mn»«mo»§#160;«/mo»«mo»§#176;«/mo»«mi mathvariant=¨normal¨»C«/mi»«mo»/«/mo»«mn»300«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/mrow»«/mstyle»«/math», the value of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»k«/mi»«/mstyle»«/math» can be calculated.


Therefore,


Since this function is only valid for altitudes below the troposphere, the restriction must be added to the function.

c.

The air temperature at a particular altitude, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»h«/mi»«/mstyle»«/math», is the temperature at sea level plus the temperature change corresponding to that altitude.

Combine the previous two functions by addition to obtain a function that could be used to predict the air temperature at any altitude below the tropopause.

d.