Answer :
To express the Fahrenheit temperature as a linear function of the Celsius temperature, we can follow these steps:
### Step 1: Understanding the Data Points
We have the following data points given:
- When the temperature is 0 degrees Celsius, the Fahrenheit temperature is 32 degrees. So we have the point (0, 32).
- When the temperature is 100 degrees Celsius, the Fahrenheit temperature is 212 degrees. So we have the point (100, 212).
### Step 2: Finding the Rate of Change (Slope)
To find the rate of change (slope), we use the formula for the slope [tex]\( m \)[/tex] of a line that passes through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Here, [tex]\((x_1, y_1) = (0, 32)\)[/tex] and [tex]\((x_2, y_2) = (100, 212)\)[/tex].
Calculating the slope:
[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 3: Forming the Linear Function
The formula for a linear function is:
[tex]\[ F(C) = mC + b \][/tex]
Where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. From the point (0, 32), we know when Celsius is 0, Fahrenheit is 32, hence [tex]\( b = 32 \)[/tex].
Therefore, the linear function is:
[tex]\[ F(C) = 1.8C + 32 \][/tex]
### Step 4: Calculating Specific Values
a. Rate of Change: From our calculation, the rate of change is [tex]\(1.8\)[/tex] Fahrenheit degrees per Celsius degree.
b. Find and Interpret [tex]\(F(27)\)[/tex]:
To find [tex]\( F(27) \)[/tex]:
[tex]\[ F(27) = 1.8 \times 27 + 32 \][/tex]
[tex]\[ F(27) = 48.6 + 32 \][/tex]
[tex]\[ F(27) = 80.6 \][/tex]
Thus, when the temperature is 27 degrees Celsius, it is approximately 80.6 degrees Fahrenheit.
c. Finding [tex]\(F(-50)\)[/tex]:
To find [tex]\( F(-50) \)[/tex]:
[tex]\[ F(-50) = 1.8 \times (-50) + 32 \][/tex]
[tex]\[ F(-50) = -90 + 32 \][/tex]
[tex]\[ F(-50) = -58 \][/tex]
Thus, when the temperature is -50 degrees Celsius, it is -58 degrees Fahrenheit.
These calculations provide a clear understanding of the Fahrenheit temperature in terms of the Celsius temperature using a linear relationship.
### Step 1: Understanding the Data Points
We have the following data points given:
- When the temperature is 0 degrees Celsius, the Fahrenheit temperature is 32 degrees. So we have the point (0, 32).
- When the temperature is 100 degrees Celsius, the Fahrenheit temperature is 212 degrees. So we have the point (100, 212).
### Step 2: Finding the Rate of Change (Slope)
To find the rate of change (slope), we use the formula for the slope [tex]\( m \)[/tex] of a line that passes through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Here, [tex]\((x_1, y_1) = (0, 32)\)[/tex] and [tex]\((x_2, y_2) = (100, 212)\)[/tex].
Calculating the slope:
[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 3: Forming the Linear Function
The formula for a linear function is:
[tex]\[ F(C) = mC + b \][/tex]
Where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. From the point (0, 32), we know when Celsius is 0, Fahrenheit is 32, hence [tex]\( b = 32 \)[/tex].
Therefore, the linear function is:
[tex]\[ F(C) = 1.8C + 32 \][/tex]
### Step 4: Calculating Specific Values
a. Rate of Change: From our calculation, the rate of change is [tex]\(1.8\)[/tex] Fahrenheit degrees per Celsius degree.
b. Find and Interpret [tex]\(F(27)\)[/tex]:
To find [tex]\( F(27) \)[/tex]:
[tex]\[ F(27) = 1.8 \times 27 + 32 \][/tex]
[tex]\[ F(27) = 48.6 + 32 \][/tex]
[tex]\[ F(27) = 80.6 \][/tex]
Thus, when the temperature is 27 degrees Celsius, it is approximately 80.6 degrees Fahrenheit.
c. Finding [tex]\(F(-50)\)[/tex]:
To find [tex]\( F(-50) \)[/tex]:
[tex]\[ F(-50) = 1.8 \times (-50) + 32 \][/tex]
[tex]\[ F(-50) = -90 + 32 \][/tex]
[tex]\[ F(-50) = -58 \][/tex]
Thus, when the temperature is -50 degrees Celsius, it is -58 degrees Fahrenheit.
These calculations provide a clear understanding of the Fahrenheit temperature in terms of the Celsius temperature using a linear relationship.
