Answer :
To solve the problem, we're looking at the relationship between the Celsius (°C) and Fahrenheit (°F) temperature scales. This relationship is linear, which means it can be represented by the equation of a straight line: [tex]\( F = m \times C + b \)[/tex], where [tex]\( F \)[/tex] is the temperature in Fahrenheit, [tex]\( C \)[/tex] is the temperature in Celsius, [tex]\( m \)[/tex] is the slope, and [tex]\( b \)[/tex] is the y-intercept.
### Step 1: Determine the linear equation
We have two points given:
- [tex]\( (C_1, F_1) = (0, 32) \)[/tex] and
- [tex]\( (C_2, F_2) = (100, 212) \)[/tex].
Using these points, we can find the slope ([tex]\( m \)[/tex]) of the line:
[tex]\[ m = \frac{F_2 - F_1}{C_2 - C_1} = \frac{212 - 32}{100 - 0} = 1.8 \][/tex]
Next, using one of the points, say [tex]\( (0, 32) \)[/tex], we can find the y-intercept ([tex]\( b \)[/tex]) by solving the equation:
[tex]\[ F = m \times C + b \][/tex]
Substituting the point:
[tex]\[ 32 = 1.8 \times 0 + b \][/tex]
Thus, [tex]\( b = 32 \)[/tex].
Therefore, the linear equation that connects Celsius and Fahrenheit is:
[tex]\[ F = 1.8 \times C + 32 \][/tex]
### Step 2: Find the Fahrenheit equivalent of 40°C
To find the Fahrenheit temperature equivalent to 40°C, substitute [tex]\( C = 40 \)[/tex] into the equation:
[tex]\[ F = 1.8 \times 40 + 32 \][/tex]
[tex]\[ F = 72 + 32 = 104 \][/tex]
So, 40°C is equivalent to 104°F.
### Step 3: Determine the temperature where °F equals °C
We need to find the temperature where Celsius and Fahrenheit readings are the same, i.e., [tex]\( F = C \)[/tex]. Substitute [tex]\( F = C \)[/tex] into the equation:
[tex]\[ C = 1.8 \times C + 32 \][/tex]
Rearranging and solving for [tex]\( C \)[/tex]:
[tex]\[ C - 1.8 \times C = 32 \][/tex]
[tex]\[ C(1 - 1.8) = 32 \][/tex]
[tex]\[ -0.8C = 32 \][/tex]
[tex]\[ C = \frac{32}{-0.8} = -40 \][/tex]
Thus, the temperature where the value of degrees Fahrenheit equals that of degrees Celsius is -40°.
In summary:
- The equation relating Celsius and Fahrenheit is [tex]\( F = 1.8 \times C + 32 \)[/tex].
- 40°C is equivalent to 104°F.
- The temperature where °F equals °C is -40°.
### Step 1: Determine the linear equation
We have two points given:
- [tex]\( (C_1, F_1) = (0, 32) \)[/tex] and
- [tex]\( (C_2, F_2) = (100, 212) \)[/tex].
Using these points, we can find the slope ([tex]\( m \)[/tex]) of the line:
[tex]\[ m = \frac{F_2 - F_1}{C_2 - C_1} = \frac{212 - 32}{100 - 0} = 1.8 \][/tex]
Next, using one of the points, say [tex]\( (0, 32) \)[/tex], we can find the y-intercept ([tex]\( b \)[/tex]) by solving the equation:
[tex]\[ F = m \times C + b \][/tex]
Substituting the point:
[tex]\[ 32 = 1.8 \times 0 + b \][/tex]
Thus, [tex]\( b = 32 \)[/tex].
Therefore, the linear equation that connects Celsius and Fahrenheit is:
[tex]\[ F = 1.8 \times C + 32 \][/tex]
### Step 2: Find the Fahrenheit equivalent of 40°C
To find the Fahrenheit temperature equivalent to 40°C, substitute [tex]\( C = 40 \)[/tex] into the equation:
[tex]\[ F = 1.8 \times 40 + 32 \][/tex]
[tex]\[ F = 72 + 32 = 104 \][/tex]
So, 40°C is equivalent to 104°F.
### Step 3: Determine the temperature where °F equals °C
We need to find the temperature where Celsius and Fahrenheit readings are the same, i.e., [tex]\( F = C \)[/tex]. Substitute [tex]\( F = C \)[/tex] into the equation:
[tex]\[ C = 1.8 \times C + 32 \][/tex]
Rearranging and solving for [tex]\( C \)[/tex]:
[tex]\[ C - 1.8 \times C = 32 \][/tex]
[tex]\[ C(1 - 1.8) = 32 \][/tex]
[tex]\[ -0.8C = 32 \][/tex]
[tex]\[ C = \frac{32}{-0.8} = -40 \][/tex]
Thus, the temperature where the value of degrees Fahrenheit equals that of degrees Celsius is -40°.
In summary:
- The equation relating Celsius and Fahrenheit is [tex]\( F = 1.8 \times C + 32 \)[/tex].
- 40°C is equivalent to 104°F.
- The temperature where °F equals °C is -40°.
