Answer :
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.
### 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.