Answer :
To express the Fahrenheit temperature as a linear function of the Celsius temperature, we need to find the relationship between Celsius and Fahrenheit.
### a. Find the rate of change
1. Identify Given Points:
- At 0 degrees Celsius, the temperature is 32 degrees Fahrenheit.
- At 100 degrees Celsius, the temperature is 212 degrees Fahrenheit.
2. Calculate the Rate of Change (Slope):
- The slope is the rate of change in Fahrenheit per degree change in Celsius.
- Using the formula for slope between two points [tex]\((C_1, F_1)\)[/tex] and [tex]\((C_2, F_2)\)[/tex]:
[tex]\[
\text{slope} = \frac{F_2 - F_1}{C_2 - C_1} = \frac{212 - 32}{100 - 0} = \frac{180}{100} = 1.8
\][/tex]
So, the rate of change is 1.8 Fahrenheit degrees per Celsius degree.
3. Determine the Linear Function:
- We use the formula for a linear function: [tex]\( F(C) = \text{slope} \times C + b \)[/tex].
4. Find the y-intercept (b):
- Substitute the known point [tex]\((C, F) = (0, 32)\)[/tex] into the equation:
[tex]\[
32 = 1.8 \times 0 + b \Rightarrow b = 32
\][/tex]
5. Write the Final Linear Function:
- Therefore, the linear function is:
[tex]\[
F(C) = 1.8C + 32
\][/tex]
### b. Find and Interpret [tex]\( F(23) \)[/tex]
1. Substitute 23 for the Celsius value in the function:
- [tex]\( F(23) = 1.8 \times 23 + 32 \)[/tex]
2. Calculate [tex]\( F(23) \)[/tex]:
- [tex]\( F(23) = 41.4 + 32 = 73.4 \)[/tex]
Interpretation: When the temperature is 23 degrees Celsius, it is 73.4 degrees Fahrenheit.
### c. Find [tex]\( F(-45) \)[/tex]
1. Substitute -45 for the Celsius value in the function:
- [tex]\( F(-45) = 1.8 \times (-45) + 32 \)[/tex]
2. Calculate [tex]\( F(-45) \)[/tex]:
- [tex]\( F(-45) = -81 + 32 = -49 \)[/tex]
So, [tex]\( F(-45) = -49 \)[/tex].
In summary:
- The Fahrenheit temperature changes by 1.8 degrees for each degree change in Celsius.
- At 23 degrees Celsius, the Fahrenheit temperature is 73.4 degrees.
- At -45 degrees Celsius, the Fahrenheit temperature is -49 degrees.
### a. Find the rate of change
1. Identify Given Points:
- At 0 degrees Celsius, the temperature is 32 degrees Fahrenheit.
- At 100 degrees Celsius, the temperature is 212 degrees Fahrenheit.
2. Calculate the Rate of Change (Slope):
- The slope is the rate of change in Fahrenheit per degree change in Celsius.
- Using the formula for slope between two points [tex]\((C_1, F_1)\)[/tex] and [tex]\((C_2, F_2)\)[/tex]:
[tex]\[
\text{slope} = \frac{F_2 - F_1}{C_2 - C_1} = \frac{212 - 32}{100 - 0} = \frac{180}{100} = 1.8
\][/tex]
So, the rate of change is 1.8 Fahrenheit degrees per Celsius degree.
3. Determine the Linear Function:
- We use the formula for a linear function: [tex]\( F(C) = \text{slope} \times C + b \)[/tex].
4. Find the y-intercept (b):
- Substitute the known point [tex]\((C, F) = (0, 32)\)[/tex] into the equation:
[tex]\[
32 = 1.8 \times 0 + b \Rightarrow b = 32
\][/tex]
5. Write the Final Linear Function:
- Therefore, the linear function is:
[tex]\[
F(C) = 1.8C + 32
\][/tex]
### b. Find and Interpret [tex]\( F(23) \)[/tex]
1. Substitute 23 for the Celsius value in the function:
- [tex]\( F(23) = 1.8 \times 23 + 32 \)[/tex]
2. Calculate [tex]\( F(23) \)[/tex]:
- [tex]\( F(23) = 41.4 + 32 = 73.4 \)[/tex]
Interpretation: When the temperature is 23 degrees Celsius, it is 73.4 degrees Fahrenheit.
### c. Find [tex]\( F(-45) \)[/tex]
1. Substitute -45 for the Celsius value in the function:
- [tex]\( F(-45) = 1.8 \times (-45) + 32 \)[/tex]
2. Calculate [tex]\( F(-45) \)[/tex]:
- [tex]\( F(-45) = -81 + 32 = -49 \)[/tex]
So, [tex]\( F(-45) = -49 \)[/tex].
In summary:
- The Fahrenheit temperature changes by 1.8 degrees for each degree change in Celsius.
- At 23 degrees Celsius, the Fahrenheit temperature is 73.4 degrees.
- At -45 degrees Celsius, the Fahrenheit temperature is -49 degrees.