Problema Solution
A station has a surface (sea-level) temperature of 10*c. If vertical soundings reveal a nearly constant environmental lapse rate of 6.5*c per kilometer and a temperature at the tropopause of -55*c, what is the height of the tropopause in kilometers? What is the tropopause height in miles (1.609 kilometers per mile)? Show your work.
Answer provided by our tutors
lets denote h = height, t = the temperature
sea level
h0 = 0 km
t0 = 10 C
after 1 km
h1 = h0 + 1
t1 = t0 - 6.5
after 2 km from the sea level
h2 = h1 + 1 = h0 + 2*1
t2 = t1 - 6.5 = t0 - 2*6.5
after n km from the sea level
hn = h0 + n*1 = 0 + n = n km
tn = t0 - n*6.5 = 10 - n*6.5
we need to find n such that the temperature is tn = - 55 that is
10 - n*6.5 = - 55
by solving the equation we find
n = 10
click here to see the step by step solution of the equation
now we can find the height: h10 = 10 km
the height of the tropopause in kilometers is 10 kilometers.
since 1 mile = 1.609 km and let h be the height in miles we have the following proportion
1 : 1.609 = h : 10
by solving we find
h = 6.215 miles
click here to see the step by step solution of the proportion
the tropopause height in miles is 6.215 miles.