As dry air moves upward it expands and in doing so cools at a rate of 1 degrees Celsius for each 100 m rise up to about 12 km

If the ground temperature is 20 degrees Celsius find an expression for the temperature T as a function of the height H
the answer is T=20-10h(h in kilometers) please show and explain steps thanks

look at a table we can set up

distance --- temp ......... by formula
0...................... 20 , T = 20 - 10(0) = 20
100 m or .1 km.. 19 , T = 20 - 10(.1) = 19
200 m or .2 km..18 , T = 20 - 10(.2) = 18
...
...
5000 m or 5 km.. -30 , T = 20 - 10(5) = -30

To find the expression for the temperature T as a function of the height H, we can start by using the information that dry air cools at a rate of 1 degree Celsius for each 100 meters rise up to about 12 km.

First, convert the height H from kilometers to meters. Since 1 kilometer is equal to 1000 meters, we have H = 1000h, where h is the height in kilometers.

Next, we can calculate the number of 100-meter intervals in the given height H. This can be done by dividing H by 100 meters.

N = H / 1000h

Since dry air cools at a rate of 1 degree Celsius for each 100 meters rise, we can multiply N by 1 to determine the decrease in temperature due to the upward movement of dry air.

ΔT = N * 1 = H / 1000h * 1 = H / 1000h

To find the temperature T as a function of the height H, we subtract the decrease in temperature ΔT from the ground temperature of 20 degrees Celsius.

T = 20 - ΔT = 20 - H / 1000h

Finally, we substitute H with 1000h to express the temperature T as a function of the height h:

T = 20 - H / 1000h = 20 - (1000h) / 1000h = 20 - 1 = 20 - h

Therefore, the expression for the temperature T as a function of the height H is T = 20 - 10h, where h is the height in kilometers.

To derive the expression for the temperature T as a function of height H, we'll follow a step-by-step process based on the given information.

Step 1: Convert the height from meters to kilometers.
Since the given rate of cooling is stated in kilometers, we need to ensure the height is also in kilometers (km):
H = h/1000

Step 2: Identify the relationship between temperature change and height change.
Given that the temperature changes at a rate of 1 degree Celsius for each 100 m rise, we can express this relationship as:
ΔT/ΔH = 1°C / 100m

Step 3: Convert the rate of temperature change to a rate in degrees Celsius per kilometer.
To ensure the rate of change is in line with the height expressed in kilometers, we need to convert the rate into degrees Celsius per kilometer. By dividing both the numerator and denominator by 100, we get:
ΔT/ΔH = 1°C / 100m = 0.01°C / m

Step 4: Integrate the rate of temperature change to find the expression for temperature T.
To obtain T as a function of H, we can integrate the rate of temperature change with respect to height H:
∫d(T)/d(H) dH = ∫(0.01°C / m) dH

Integrating both sides gives us:
∫d(T) = 0.01°C ∫d(H)

Simplifying further:
(T + C1) = 0.01°C H + C2

Since we are given that the ground temperature is 20 degrees Celsius, we substitute T = 20°C and H = 0 into the equation to find the integration constants C1 and C2:

(20°C + C1) = 0.01°C (0km) + C2
C1 = 20°C

Thus, the equation becomes:
T + 20°C = 0.01°C H + C2

Step 5: Find the integration constant C2 using the height limit of 12 km.
Since the given information states that the rate of cooling holds up to about 12 km, we substitute T = -10°C (12 km) and H = 12 into the equation:

T + 20°C = 0.01°C H + C2
-10°C + 20°C = 0.01°C (12 km) + C2
10°C = 0.12°C + C2
C2 = 10°C - 0.12°C
C2 = 9.88°C

Step 6: Substitute the integration constants back into the equation.
Now we can substitute C1 = 20°C and C2 = 9.88°C into the equation:

T + 20°C = 0.01°C H + 9.88°C

Finally, rearranging the equation to solve for T:
T = 0.01°C H + 9.88°C - 20°C
T = 0.01°C H - 10.12°C

So, the expression for temperature T as a function of height H is:
T = 0.01°C H - 10.12°C

This equation describes how temperature changes with height based on the given rate of cooling.