In the Fahrenheit temperature scale, water freezes at 32°F and boils at 212°F. In the Celsius scale, water freezes at 0°C and boils at 100°C. Given that the Fahrenheit temperature F and the Celsius temperature C are related by a linear equation, find F in terms of C.
The Correct Answer and Explanation is :
To find the relationship between Fahrenheit (F) and Celsius (C), we use the fact that the freezing and boiling points of water are the same in both scales, just at different temperatures. The linear equation that relates these two temperature scales can be derived step by step.
Step 1: Understand the freezing and boiling points
- Water freezes at 32°F and 0°C.
- Water boils at 212°F and 100°C.
Step 2: Derive the equation
Since the relationship between Fahrenheit and Celsius is linear, the equation can be written as:
[
F = mC + b
]
where ( m ) is the slope (the rate at which Fahrenheit changes with respect to Celsius), and ( b ) is the y-intercept (the Fahrenheit temperature when Celsius is 0).
We already know two points: when ( C = 0 ), ( F = 32 ), and when ( C = 100 ), ( F = 212 ). Using these two points, we can find ( m ) and ( b ).
Step 3: Find the slope ( m )
The slope of the line is the change in Fahrenheit divided by the change in Celsius:
[
m = \frac{F_2 – F_1}{C_2 – C_1} = \frac{212 – 32}{100 – 0} = \frac{180}{100} = 1.8
]
Step 4: Find the y-intercept ( b )
Now, substitute one of the points into the equation to find ( b ). Using ( C = 0 ) and ( F = 32 ):
[
32 = 1.8(0) + b \quad \Rightarrow \quad b = 32
]
Step 5: Write the final equation
Thus, the equation that relates Fahrenheit and Celsius is:
[
F = 1.8C + 32
]
Explanation
This equation tells us how to convert a temperature from Celsius to Fahrenheit. The factor 1.8 represents the rate at which Fahrenheit increases for each degree increase in Celsius. The constant 32 is the offset needed to adjust for the different starting points of the two scales.
For example, if the temperature is 0°C, it is 32°F. If the temperature is 100°C, it is 212°F. By using this linear equation, you can easily convert any temperature between the two scales.