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.