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When the temperature is 0 degrees Celsius, the Fahrenheit temperature is 32. When the Celsius temperature is 100, the corresponding Fahrenheit temperature is 212.

Express the Fahrenheit temperature as a linear function of [tex]C[/tex], the Celsius temperature, [tex]F(C)[/tex].

\[ F(C) = \square \]

a. Find the rate of change of Fahrenheit temperature for each unit change in Celsius temperature.

\[ \square \] Fahrenheit degrees per Celsius degree

b. Find and interpret [tex]F(22)[/tex]. Round to 1 decimal place.

At [tex]\square[/tex] degrees Celsius, it is [tex]\square[/tex] degrees Fahrenheit.

c. [tex]F(-35) = \square[/tex]

Answer :

To express the Fahrenheit temperature as a linear function of the Celsius temperature, [tex]\( F(C) \)[/tex], we'll follow these steps:

### Step 1: Establish the Linear Relationship

We know two key points that relate Celsius and Fahrenheit temperatures:
- At 0 degrees Celsius, the temperature in Fahrenheit is 32 degrees.
- At 100 degrees Celsius, the temperature in Fahrenheit is 212 degrees.

### Step 2: Calculate the Slope

The formula for the slope ([tex]\(m\)[/tex]) between two points [tex]\((C_1, F_1)\)[/tex] and [tex]\((C_2, F_2)\)[/tex] is:

[tex]\[
m = \frac{F_2 - F_1}{C_2 - C_1}
\][/tex]

Substituting our known values:
- [tex]\(C_1 = 0\)[/tex], [tex]\(F_1 = 32\)[/tex]
- [tex]\(C_2 = 100\)[/tex], [tex]\(F_2 = 212\)[/tex]

[tex]\[
m = \frac{212 - 32}{100 - 0} = \frac{180}{100} = 1.8
\][/tex]

So, the rate of change is 1.8 Fahrenheit degrees per Celsius degree.

### Step 3: Write the Linear Function

The linear equation can be written as:

[tex]\[
F(C) = mC + b
\][/tex]

Where [tex]\(b\)[/tex] is the y-intercept. Since we know when [tex]\(C = 0\)[/tex], [tex]\(F = 32\)[/tex], the y-intercept [tex]\(b\)[/tex] is 32.

Thus, the function is:

[tex]\[
F(C) = 1.8C + 32
\][/tex]

### Step 4: Interpreting [tex]\(F(22)\)[/tex]

To find [tex]\(F(22)\)[/tex], substitute 22 into the function:

[tex]\[
F(22) = 1.8 \times 22 + 32 = 39.6 + 32 = 71.6
\][/tex]

So, at 22 degrees Celsius, it is 71.6 degrees Fahrenheit.

### Step 5: Calculate [tex]\(F(-35)\)[/tex]

To find [tex]\(F(-35)\)[/tex], substitute -35 into the function:

[tex]\[
F(-35) = 1.8 \times (-35) + 32 = -63 + 32 = -31
\][/tex]

Therefore, at -35 degrees Celsius, it is -31 degrees Fahrenheit.

These calculations provide the linear relationship between Celsius and Fahrenheit temperatures and specific evaluations for given Celsius values.

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