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.
### 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.
