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

The air temperature at the tropopause is - 79 °C

Step-by-step 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