As dry air moves upward , it expands and cools. If the temperature on the ground is `20^@` C and the temperature at a height of 1 km is `10^@` C, which of the following linear models best describes the temperature t in degrees C at a height of h km above the ground ?

Options

A. `t=-1/10h+20`
B. `t=1/10h+20`
C. t=-10h+20
D. t=10h+20

Answer

To solve the problem step by step, we need to find a linear model that describes the relationship between temperature \( T \) (in degrees Celsius) and height \( h \) (in kilometers) based on the given data.

### Step 1: Understand the given information
We know:
- The temperature at ground level (when \( h = 0 \)) is \( T = 20^\circ C \).
- The temperature at a height of 1 km (when \( h = 1 \)) is \( T = 10^\circ C \).

### Step 2: Set up the linear equation
We assume a linear relationship between temperature \( T \) and height \( h \):
\[ T = a \cdot h + b \]
where \( a \) and \( b \) are constants we need to determine.

### Step 3: Use the first data point
From the first data point, when \( h = 0 \):
\[ T = 20 \implies 20 = a \cdot 0 + b \implies b = 20 \]

### Step 4: Substitute \( b \) into the equation
Now substituting \( b \) back into the linear equation:
\[ T = a \cdot h + 20 \]

### Step 5: Use the second data point
From the second data point, when \( h = 1 \):
\[ T = 10 \implies 10 = a \cdot 1 + 20 \]
This simplifies to:
\[ 10 = a + 20 \]
Subtracting 20 from both sides gives:
\[ a = 10 - 20 = -10 \]

### Step 6: Write the final equation
Now substituting the value of \( a \) back into the equation:
\[ T = -10 \cdot h + 20 \]

### Step 7: Identify the correct model
The linear model that describes the temperature \( T \) at height \( h \) is:
\[ T = -10h + 20 \]
This matches with option 3 from the provided choices.

### Final Answer
Thus, the correct linear model is:
\[ T = -10h + 20 \]

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