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

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

[tex]F(C) = \square[/tex]

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

Rate: [tex]\square[/tex] Fahrenheit degrees per Celsius degree

3. 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.

4. Calculate [tex]F(-35)[/tex].

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

Let's solve the problem step by step.

### a. Finding the rate of change of Fahrenheit temperature per degree Celsius:

1. We are given two points on the Celsius-Fahrenheit scale:
- When Celsius is 0, Fahrenheit is 32.
- When Celsius is 100, Fahrenheit is 212.

2. To find the rate of change, or the slope [tex]$ m $[/tex], of the linear function, we use the formula for the slope between two points [tex]$(x_1, y_1)$[/tex] and [tex]$(x_2, y_2)$[/tex]:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Here, [tex]$x_1 = 0$[/tex], [tex]$y_1 = 32$[/tex], [tex]$x_2 = 100$[/tex], and [tex]$y_2 = 212$[/tex].

3. Plugging the values into the formula, we get:
[tex]\[
m = \frac{212 - 32}{100 - 0} = \frac{180}{100} = 1.8
\][/tex]

The rate of change is 1.8 Fahrenheit degrees per Celsius degree.

### b. Finding and interpreting [tex]$ F(22) $[/tex]:

1. To express Fahrenheit [tex]$ F $[/tex] as a function of Celsius [tex]$ C $[/tex], use:
[tex]\[
F(C) = mC + b
\][/tex]
where [tex]$ m = 1.8 $[/tex] and [tex]$ b = 32 $[/tex] (since [tex]$ F(0) = 32 $[/tex]).

2. Substitute 22 for [tex]$ C $[/tex] in the function:
[tex]\[
F(22) = 1.8 \times 22 + 32
\][/tex]

3. Calculating the value:
[tex]\[
F(22) = 39.6 + 32 = 71.6
\][/tex]

When the temperature is 22 degrees Celsius, the corresponding Fahrenheit temperature is 71.6 degrees.

### c. Finding [tex]$ F(-35) $[/tex]:

1. Again using the linear function [tex]$ F(C) = 1.8C + 32 $[/tex], substitute [tex]$-35$[/tex] for [tex]$ C $[/tex]:
[tex]\[
F(-35) = 1.8 \times (-35) + 32
\][/tex]

2. Calculating the value:
[tex]\[
F(-35) = -63 + 32 = -31
\][/tex]

So, at [tex]$-35$[/tex] degrees Celsius, the Fahrenheit temperature is [tex]$-31$[/tex] degrees.