Thank you for visiting When the temperature is 0 degrees Celsius the Fahrenheit temperature is 32 When the Celsius temperature is 100 the corresponding Fahrenheit temperature is 212 1. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
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.
Thank you for reading the article When the temperature is 0 degrees Celsius the Fahrenheit temperature is 32 When the Celsius temperature is 100 the corresponding Fahrenheit temperature is 212 1. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
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