Answer :
To express the Fahrenheit temperature as a linear function of the Celsius temperature [tex]\( C \)[/tex], and to answer the questions provided, we can follow these steps:
1. Creating the Linear Function [tex]\( F(C) \)[/tex]:
We know that the relationship between Celsius and Fahrenheit temperatures is linear. We have two key data points:
- When Celsius is 0 degrees, Fahrenheit is 32 degrees.
- When Celsius is 100 degrees, Fahrenheit is 212 degrees.
Using these points, we can determine the formula for the linear function. The general form of a linear equation is:
[tex]\[
F(C) = m \times C + b
\][/tex]
where [tex]\( m \)[/tex] is the slope, and [tex]\( b \)[/tex] is the y-intercept.
2. Calculating the Slope [tex]\( m \)[/tex]:
The slope [tex]\( m \)[/tex] can be calculated using the formula for the slope between two points:
[tex]\[
m = \frac{F_2 - F_1}{C_2 - C_1} = \frac{212 - 32}{100 - 0} = \frac{180}{100} = 1.8
\][/tex]
3. Finding the Linear Function [tex]\( F(C) \)[/tex]:
We already know that when [tex]\( C = 0 \)[/tex], [tex]\( F = 32 \)[/tex], giving us the y-intercept [tex]\( b = 32 \)[/tex]. Therefore, the function is:
[tex]\[
F(C) = 1.8 \times C + 32
\][/tex]
4. Answering the Questions:
a. Rate of Change:
The rate of change of Fahrenheit temperature for each unit change in Celsius is the slope [tex]\( m \)[/tex], which is [tex]\( 1.8 \)[/tex] Fahrenheit degrees per Celsius degree.
b. Finding and Interpreting [tex]\( F(20) \)[/tex]:
To find the Fahrenheit temperature when Celsius is 20 degrees, substitute [tex]\( 20 \)[/tex] for [tex]\( C \)[/tex] in the linear function:
[tex]\[
F(20) = 1.8 \times 20 + 32 = 36 + 32 = 68.0
\][/tex]
So, at 20 degrees Celsius, it is 68.0 degrees Fahrenheit.
c. Finding [tex]\( F(-45) \)[/tex]:
To find the Fahrenheit temperature when Celsius is -45 degrees, substitute [tex]\( -45 \)[/tex] for [tex]\( C \)[/tex] in the linear function:
[tex]\[
F(-45) = 1.8 \times (-45) + 32 = -81 + 32 = -49.0
\][/tex]
Therefore, at -45 degrees Celsius, it is -49.0 degrees Fahrenheit.
This step-by-step guide shows how we derive the linear relationship between Celsius and Fahrenheit and use it to solve the given problems accurately.
1. Creating the Linear Function [tex]\( F(C) \)[/tex]:
We know that the relationship between Celsius and Fahrenheit temperatures is linear. We have two key data points:
- When Celsius is 0 degrees, Fahrenheit is 32 degrees.
- When Celsius is 100 degrees, Fahrenheit is 212 degrees.
Using these points, we can determine the formula for the linear function. The general form of a linear equation is:
[tex]\[
F(C) = m \times C + b
\][/tex]
where [tex]\( m \)[/tex] is the slope, and [tex]\( b \)[/tex] is the y-intercept.
2. Calculating the Slope [tex]\( m \)[/tex]:
The slope [tex]\( m \)[/tex] can be calculated using the formula for the slope between two points:
[tex]\[
m = \frac{F_2 - F_1}{C_2 - C_1} = \frac{212 - 32}{100 - 0} = \frac{180}{100} = 1.8
\][/tex]
3. Finding the Linear Function [tex]\( F(C) \)[/tex]:
We already know that when [tex]\( C = 0 \)[/tex], [tex]\( F = 32 \)[/tex], giving us the y-intercept [tex]\( b = 32 \)[/tex]. Therefore, the function is:
[tex]\[
F(C) = 1.8 \times C + 32
\][/tex]
4. Answering the Questions:
a. Rate of Change:
The rate of change of Fahrenheit temperature for each unit change in Celsius is the slope [tex]\( m \)[/tex], which is [tex]\( 1.8 \)[/tex] Fahrenheit degrees per Celsius degree.
b. Finding and Interpreting [tex]\( F(20) \)[/tex]:
To find the Fahrenheit temperature when Celsius is 20 degrees, substitute [tex]\( 20 \)[/tex] for [tex]\( C \)[/tex] in the linear function:
[tex]\[
F(20) = 1.8 \times 20 + 32 = 36 + 32 = 68.0
\][/tex]
So, at 20 degrees Celsius, it is 68.0 degrees Fahrenheit.
c. Finding [tex]\( F(-45) \)[/tex]:
To find the Fahrenheit temperature when Celsius is -45 degrees, substitute [tex]\( -45 \)[/tex] for [tex]\( C \)[/tex] in the linear function:
[tex]\[
F(-45) = 1.8 \times (-45) + 32 = -81 + 32 = -49.0
\][/tex]
Therefore, at -45 degrees Celsius, it is -49.0 degrees Fahrenheit.
This step-by-step guide shows how we derive the linear relationship between Celsius and Fahrenheit and use it to solve the given problems accurately.
