The mesosphere is the layer of Earth’s atmosphere between 50 kilometers and 85 kilometers above Earth’s surface. At a distance of 50 kilometers from Earth’s surface, the temperature in the mesosphere is `-5^(@)` Celsius, and at a distance of 80 kilometers from Earth’s surface, the temperature in the mesosphere is `-80^(@)` Celsius. For every additional 10 kilometers from Earth’s surface, the temperature in the mesosphere decreases by `k^(@)` Celsius, where k is a constant. What is the value of k ?

Explanation

To find the value of \( k \), we can follow these steps:

### Step 1: Identify the temperature at two distances
We know:
- At a distance of \( 50 \) km, the temperature \( T_1 = -5^\circ C \).
- At a distance of \( 80 \) km, the temperature \( T_2 = -80^\circ C \).

### Step 2: Calculate the change in temperature
The change in temperature (\( \Delta T \)) can be calculated as:
\[ \Delta T = T_2 - T_1 = -80 - (-5) = -80 + 5 = -75^\circ C \]

### Step 3: Calculate the change in distance
The change in distance (\( \Delta D \)) is:
\[ \Delta D = D_2 - D_1 = 80 - 50 = 30 \text{ km} \]

### Step 4: Calculate the rate of change of temperature per kilometer
The rate of change of temperature per kilometer (slope) is given by:
\[ \text{slope} = \frac{\Delta T}{\Delta D} = \frac{-75}{30} = -2.5^\circ C/\text{km} \]

### Step 5: Determine the value of \( k \)
Since the temperature decreases by \( k \) degrees Celsius for every additional \( 10 \) kilometers, we can find \( k \) by multiplying the rate of change by \( 10 \):
\[ k = 10 \times (-2.5) = -25^\circ C \]

### Step 6: Finalize the answer
Thus, the value of \( k \) is:
\[ k = 25^\circ C \]