Question

On the same spring day, a station near the equator has a surface temperature of 25°C, 15°C higher than the middle-latitude city mentioned in part a. Vertical soundings reveal an environmental lapse rate of 6.5°C per kilometer and indicate that the tropopause is encountered at 16 km. What is the air temperature at the tropopause?

Answer 1

Answer:

The air temperature at the tropopause is - 79 °C

Explanation:

We know that a station near the equator has a surface temperature of 25°C

Vertical soundings reveal an environmental lapse rate of 6.5 °C per kilometer.

The tropopause is encountered at 16 km.

In order to find the air temperature at the tropopause we are going to deduce a linear function for the temperature at the tropopause.

This linear function will have the following structure :

$f(x)=ax+b$

Where ''$a$'' and ''$b$'' are real numbers.

Let's write $T(x)$ to denote the temperature '' T '' in function of the distance

'' x '' ⇒

$T(x)=ax+b$

We can reorder the function as :

$T(x)=b+ax$ (I)

Now, at the surface the value of ''$x$'' is 0 km and the temperature is 25°C so in the function (I) we write :

$T(0)=25=b+a(0)$ ⇒ $b=25$ ⇒

$T(x)=25+ax$ (II)

In (II) the value of ''$a$'' represents the change in temperature per kilometer.

Because the temperature decrease with the height this number will be negative and also a data from the question ⇒

$T(x)=25-(6.5)x$ (III)

In (III) we deduced the linear equation. The last step is to replace by $x=16$ in (III) ⇒

$T(16)=25-(6.5)(16)=-79$

The air temperature at the tropopause is - 79 °C