Answer :
Let's work through this step-by-step:
We need to express the Fahrenheit temperature [tex]\( F \)[/tex] as a linear function of the Celsius temperature [tex]\( C \)[/tex].
Step 1: Determine the slope (rate of change)
We have two points from the problem:
- When Celsius is 0, Fahrenheit is 32: [tex]\((0, 32)\)[/tex]
- When Celsius is 100, Fahrenheit is 212: [tex]\((100, 212)\)[/tex]
The slope (rate of change) of a linear function is calculated as:
[tex]\[ \text{slope} = \frac{\text{change in Fahrenheit}}{\text{change in Celsius}} = \frac{F_2 - F_1}{C_2 - C_1} = \frac{212 - 32}{100 - 0} = \frac{180}{100} = 1.8 \][/tex]
So, the rate of change is 1.8 Fahrenheit degrees per Celsius degree.
Step 2: Write the linear function
The general form of a linear equation is:
[tex]\[ F(C) = mC + b \][/tex]
Where [tex]\( m \)[/tex] is the slope, which we found to be 1.8, and [tex]\( b \)[/tex] is the y-intercept. Since [tex]\( F(0) = 32 \)[/tex], the intercept [tex]\( b \)[/tex] is 32.
Thus, the function is:
[tex]\[ F(C) = 1.8C + 32 \][/tex]
Step 3: Calculate [tex]\( F(25) \)[/tex]
Now, plug [tex]\( C = 25 \)[/tex] into the linear function:
[tex]\[ F(25) = 1.8 \times 25 + 32 \][/tex]
[tex]\[ F(25) = 45 + 32 = 77 \][/tex]
So, when the Celsius temperature is 25 degrees, the Fahrenheit temperature is 77.0 degrees.
Step 4: Calculate [tex]\( F(-50) \)[/tex]
Plug [tex]\( C = -50 \)[/tex] into the linear function:
[tex]\[ F(-50) = 1.8 \times (-50) + 32 \][/tex]
[tex]\[ F(-50) = -90 + 32 = -58 \][/tex]
So, when the Celsius temperature is -50 degrees, the Fahrenheit temperature is -58.0 degrees.
In summary:
- The rate of change is 1.8 Fahrenheit degrees per Celsius degree.
- [tex]\( F(25) = 77.0 \)[/tex] Fahrenheit degrees.
- [tex]\( F(-50) = -58.0 \)[/tex] Fahrenheit degrees.
We need to express the Fahrenheit temperature [tex]\( F \)[/tex] as a linear function of the Celsius temperature [tex]\( C \)[/tex].
Step 1: Determine the slope (rate of change)
We have two points from the problem:
- When Celsius is 0, Fahrenheit is 32: [tex]\((0, 32)\)[/tex]
- When Celsius is 100, Fahrenheit is 212: [tex]\((100, 212)\)[/tex]
The slope (rate of change) of a linear function is calculated as:
[tex]\[ \text{slope} = \frac{\text{change in Fahrenheit}}{\text{change in Celsius}} = \frac{F_2 - F_1}{C_2 - C_1} = \frac{212 - 32}{100 - 0} = \frac{180}{100} = 1.8 \][/tex]
So, the rate of change is 1.8 Fahrenheit degrees per Celsius degree.
Step 2: Write the linear function
The general form of a linear equation is:
[tex]\[ F(C) = mC + b \][/tex]
Where [tex]\( m \)[/tex] is the slope, which we found to be 1.8, and [tex]\( b \)[/tex] is the y-intercept. Since [tex]\( F(0) = 32 \)[/tex], the intercept [tex]\( b \)[/tex] is 32.
Thus, the function is:
[tex]\[ F(C) = 1.8C + 32 \][/tex]
Step 3: Calculate [tex]\( F(25) \)[/tex]
Now, plug [tex]\( C = 25 \)[/tex] into the linear function:
[tex]\[ F(25) = 1.8 \times 25 + 32 \][/tex]
[tex]\[ F(25) = 45 + 32 = 77 \][/tex]
So, when the Celsius temperature is 25 degrees, the Fahrenheit temperature is 77.0 degrees.
Step 4: Calculate [tex]\( F(-50) \)[/tex]
Plug [tex]\( C = -50 \)[/tex] into the linear function:
[tex]\[ F(-50) = 1.8 \times (-50) + 32 \][/tex]
[tex]\[ F(-50) = -90 + 32 = -58 \][/tex]
So, when the Celsius temperature is -50 degrees, the Fahrenheit temperature is -58.0 degrees.
In summary:
- The rate of change is 1.8 Fahrenheit degrees per Celsius degree.
- [tex]\( F(25) = 77.0 \)[/tex] Fahrenheit degrees.
- [tex]\( F(-50) = -58.0 \)[/tex] Fahrenheit degrees.
