Thank you for visiting 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. 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 :
Sure! Let's break down the solution step-by-step.
We need to express the Fahrenheit temperature [tex]\( F(C) \)[/tex] as a linear function of the Celsius temperature [tex]\( C \)[/tex]. We're given two points:
- When the Celsius temperature is 0 degrees, the Fahrenheit temperature is 32 degrees.
So, the point is [tex]\((0, 32)\)[/tex].
- When the Celsius temperature is 100 degrees, the Fahrenheit temperature is 212 degrees.
So, the point is [tex]\((100, 212)\)[/tex].
### Step a: Find the rate of change of Fahrenheit temperature for each unit change in Celsius temperature.
The rate of change in the context of a linear function is the slope. The slope [tex]\( m \)[/tex] can be found using the two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] using the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substitute the points [tex]\((0, 32)\)[/tex] and [tex]\((100, 212)\)[/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 b: Find and interpret [tex]\( F(21) \)[/tex].
The equation for the linear function is of the form [tex]\( F(C) = mC + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept, which we can determine from one of the initial given points.
Using the point [tex]\((0, 32)\)[/tex], [tex]\( b = 32 \)[/tex].
Therefore, the linear function becomes:
[tex]\[
F(C) = 1.8C + 32
\][/tex]
To find [tex]\( F(21) \)[/tex], substitute [tex]\( C = 21 \)[/tex] into the equation:
[tex]\[
F(21) = 1.8 \times 21 + 32 = 37.8 + 32 = 69.8
\][/tex]
So, at 21 degrees Celsius, it is 69.8 degrees Fahrenheit.
### Step c: [tex]\( F(-30) \)[/tex]
To find [tex]\( F(-30) \)[/tex], substitute [tex]\( C = -30 \)[/tex] into the equation:
[tex]\[
F(-30) = 1.8 \times (-30) + 32 = -54 + 32 = -22.0
\][/tex]
So, at -30 degrees Celsius, it is -22.0 degrees Fahrenheit.
Overall, we have calculated:
- The rate of change is 1.8 Fahrenheit degrees per Celsius degree.
- At 21 degrees Celsius, the Fahrenheit temperature is 69.8 degrees.
- At -30 degrees Celsius, the Fahrenheit temperature is -22.0 degrees.
We need to express the Fahrenheit temperature [tex]\( F(C) \)[/tex] as a linear function of the Celsius temperature [tex]\( C \)[/tex]. We're given two points:
- When the Celsius temperature is 0 degrees, the Fahrenheit temperature is 32 degrees.
So, the point is [tex]\((0, 32)\)[/tex].
- When the Celsius temperature is 100 degrees, the Fahrenheit temperature is 212 degrees.
So, the point is [tex]\((100, 212)\)[/tex].
### Step a: Find the rate of change of Fahrenheit temperature for each unit change in Celsius temperature.
The rate of change in the context of a linear function is the slope. The slope [tex]\( m \)[/tex] can be found using the two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] using the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substitute the points [tex]\((0, 32)\)[/tex] and [tex]\((100, 212)\)[/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 b: Find and interpret [tex]\( F(21) \)[/tex].
The equation for the linear function is of the form [tex]\( F(C) = mC + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept, which we can determine from one of the initial given points.
Using the point [tex]\((0, 32)\)[/tex], [tex]\( b = 32 \)[/tex].
Therefore, the linear function becomes:
[tex]\[
F(C) = 1.8C + 32
\][/tex]
To find [tex]\( F(21) \)[/tex], substitute [tex]\( C = 21 \)[/tex] into the equation:
[tex]\[
F(21) = 1.8 \times 21 + 32 = 37.8 + 32 = 69.8
\][/tex]
So, at 21 degrees Celsius, it is 69.8 degrees Fahrenheit.
### Step c: [tex]\( F(-30) \)[/tex]
To find [tex]\( F(-30) \)[/tex], substitute [tex]\( C = -30 \)[/tex] into the equation:
[tex]\[
F(-30) = 1.8 \times (-30) + 32 = -54 + 32 = -22.0
\][/tex]
So, at -30 degrees Celsius, it is -22.0 degrees Fahrenheit.
Overall, we have calculated:
- The rate of change is 1.8 Fahrenheit degrees per Celsius degree.
- At 21 degrees Celsius, the Fahrenheit temperature is 69.8 degrees.
- At -30 degrees Celsius, the Fahrenheit temperature is -22.0 degrees.
Thank you for reading the article 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. 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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