Answer :
We are given that when the Celsius temperature is [tex]$0^\circ$[/tex], the Fahrenheit temperature is [tex]$32^\circ$[/tex], and when the Celsius temperature is [tex]$100^\circ$[/tex], the Fahrenheit temperature is [tex]$212^\circ$[/tex]. We want to express the Fahrenheit temperature as a linear function of the Celsius temperature, [tex]$F(C)$[/tex].
To write this as a linear function, we use the form
[tex]$$
F(C) = mC + b,
$$[/tex]
where [tex]$m$[/tex] is the rate of change (slope) and [tex]$b$[/tex] is the [tex]$y$[/tex]-intercept.
Step 1. Find the slope [tex]$m$[/tex].
The slope is given by
[tex]$$
m = \frac{F(100) - F(0)}{100 - 0} = \frac{212 - 32}{100} = \frac{180}{100} = 1.8.
$$[/tex]
So, the rate of change is [tex]$1.8$[/tex] Fahrenheit degrees per Celsius degree.
Step 2. Find the [tex]$y$[/tex]-intercept [tex]$b$[/tex].
Since [tex]$F(0) = 32$[/tex], it follows directly that
[tex]$$
b = 32.
$$[/tex]
Step 3. Write the linear function.
Substituting [tex]$m = 1.8$[/tex] and [tex]$b = 32$[/tex] into the linear formula gives
[tex]$$
F(C) = 1.8C + 32.
$$[/tex]
Step 4. Evaluate [tex]$F(27)$[/tex].
Substitute [tex]$C = 27$[/tex] into the function:
[tex]$$
F(27) = 1.8 \times 27 + 32.
$$[/tex]
Calculating this,
[tex]$$
1.8 \times 27 = 48.6,
$$[/tex]
so
[tex]$$
F(27) = 48.6 + 32 = 80.6.
$$[/tex]
Rounded to one decimal place, [tex]$F(27)$[/tex] is [tex]$80.6$[/tex]. This means that when the Celsius temperature is [tex]$27^\circ$[/tex], the Fahrenheit temperature is [tex]$80.6^\circ$[/tex].
Step 5. Evaluate [tex]$F(-30)$[/tex].
Substitute [tex]$C = -30$[/tex] into the function:
[tex]$$
F(-30) = 1.8 \times (-30) + 32.
$$[/tex]
Calculating,
[tex]$$
1.8 \times (-30) = -54,
$$[/tex]
so
[tex]$$
F(-30) = -54 + 32 = -22.
$$[/tex]
Thus, [tex]$F(-30) = -22^\circ$[/tex] Fahrenheit.
Summarizing the results:
a. The rate of change is [tex]$1.8$[/tex] Fahrenheit degrees per Celsius degree.
b. The function is [tex]$F(C) = 1.8C + 32$[/tex], and [tex]$F(27) = 80.6^\circ$[/tex] Fahrenheit. In other words, at [tex]$27^\circ$[/tex] Celsius, it is [tex]$80.6^\circ$[/tex] Fahrenheit.
c. [tex]$F(-30) = -22^\circ$[/tex] Fahrenheit.
To write this as a linear function, we use the form
[tex]$$
F(C) = mC + b,
$$[/tex]
where [tex]$m$[/tex] is the rate of change (slope) and [tex]$b$[/tex] is the [tex]$y$[/tex]-intercept.
Step 1. Find the slope [tex]$m$[/tex].
The slope is given by
[tex]$$
m = \frac{F(100) - F(0)}{100 - 0} = \frac{212 - 32}{100} = \frac{180}{100} = 1.8.
$$[/tex]
So, the rate of change is [tex]$1.8$[/tex] Fahrenheit degrees per Celsius degree.
Step 2. Find the [tex]$y$[/tex]-intercept [tex]$b$[/tex].
Since [tex]$F(0) = 32$[/tex], it follows directly that
[tex]$$
b = 32.
$$[/tex]
Step 3. Write the linear function.
Substituting [tex]$m = 1.8$[/tex] and [tex]$b = 32$[/tex] into the linear formula gives
[tex]$$
F(C) = 1.8C + 32.
$$[/tex]
Step 4. Evaluate [tex]$F(27)$[/tex].
Substitute [tex]$C = 27$[/tex] into the function:
[tex]$$
F(27) = 1.8 \times 27 + 32.
$$[/tex]
Calculating this,
[tex]$$
1.8 \times 27 = 48.6,
$$[/tex]
so
[tex]$$
F(27) = 48.6 + 32 = 80.6.
$$[/tex]
Rounded to one decimal place, [tex]$F(27)$[/tex] is [tex]$80.6$[/tex]. This means that when the Celsius temperature is [tex]$27^\circ$[/tex], the Fahrenheit temperature is [tex]$80.6^\circ$[/tex].
Step 5. Evaluate [tex]$F(-30)$[/tex].
Substitute [tex]$C = -30$[/tex] into the function:
[tex]$$
F(-30) = 1.8 \times (-30) + 32.
$$[/tex]
Calculating,
[tex]$$
1.8 \times (-30) = -54,
$$[/tex]
so
[tex]$$
F(-30) = -54 + 32 = -22.
$$[/tex]
Thus, [tex]$F(-30) = -22^\circ$[/tex] Fahrenheit.
Summarizing the results:
a. The rate of change is [tex]$1.8$[/tex] Fahrenheit degrees per Celsius degree.
b. The function is [tex]$F(C) = 1.8C + 32$[/tex], and [tex]$F(27) = 80.6^\circ$[/tex] Fahrenheit. In other words, at [tex]$27^\circ$[/tex] Celsius, it is [tex]$80.6^\circ$[/tex] Fahrenheit.
c. [tex]$F(-30) = -22^\circ$[/tex] Fahrenheit.