Answer :
We are given that when the Celsius temperature is 0, the Fahrenheit temperature is 32, and when the Celsius temperature is 100, the Fahrenheit temperature is 212. We wish to express the Fahrenheit temperature as a linear function of Celsius temperature, that is, in the form
[tex]$$
F(C) = aC + b.
$$[/tex]
Step 1. Find the constant [tex]$b$[/tex] by using the point where [tex]$C = 0$[/tex].
Since [tex]$F(0) = 32$[/tex], we have
[tex]$$
F(0) = a(0) + b = 32 \quad \Longrightarrow \quad b = 32.
$$[/tex]
Step 2. Find the slope [tex]$a$[/tex] using the second given point.
When [tex]$C = 100$[/tex], [tex]$F(100) = 212$[/tex]. Substitute into the linear function:
[tex]$$
F(100) = a(100) + 32 = 212.
$$[/tex]
Step 3. Solve for [tex]$a$[/tex].
Subtract 32 from both sides:
[tex]$$
100a = 212 - 32 = 180.
$$[/tex]
Divide both sides by 100:
[tex]$$
a = \frac{180}{100} = 1.8.
$$[/tex]
Thus, the linear function that expresses Fahrenheit in terms of Celsius is:
[tex]$$
F(C) = 1.8C + 32.
$$[/tex]
---
Part (a): Find the rate of change of Fahrenheit temperature for each unit change in Celsius.
The rate of change is the slope [tex]$a$[/tex], which is
[tex]$$
1.8 \text{ Fahrenheit degrees per Celsius degree}.
$$[/tex]
---
Part (b): Find and interpret [tex]$F(21)$[/tex].
Substitute [tex]$C = 21$[/tex] into the linear function:
[tex]$$
F(21) = 1.8(21) + 32.
$$[/tex]
Calculating,
[tex]$$
1.8 \times 21 = 37.8,
$$[/tex]
so
[tex]$$
F(21) = 37.8 + 32 = 69.8.
$$[/tex]
This means that when the Celsius temperature is [tex]$21^\circ\text{C}$[/tex], the Fahrenheit temperature is [tex]$69.8^\circ\text{F}$[/tex].
---
Part (c): Find [tex]$F(-30)$[/tex].
Substitute [tex]$C = -30$[/tex] into the linear function:
[tex]$$
F(-30) = 1.8(-30) + 32.
$$[/tex]
Calculating,
[tex]$$
1.8 \times (-30) = -54,
$$[/tex]
so
[tex]$$
F(-30) = -54 + 32 = -22.
$$[/tex]
Thus, when the Celsius temperature is [tex]$-30^\circ\text{C}$[/tex], the Fahrenheit temperature is [tex]$-22^\circ\text{F}$[/tex].
---
Summary of Answers:
1. The linear function is:
[tex]$$
F(C) = 1.8C + 32.
$$[/tex]
2. The rate of change is [tex]$1.8$[/tex] Fahrenheit degrees per Celsius degree.
3. [tex]$F(21) = 69.8$[/tex], meaning that [tex]$21^\circ\text{C}$[/tex] is equivalent to [tex]$69.8^\circ\text{F}$[/tex].
4. [tex]$F(-30) = -22$[/tex].
[tex]$$
F(C) = aC + b.
$$[/tex]
Step 1. Find the constant [tex]$b$[/tex] by using the point where [tex]$C = 0$[/tex].
Since [tex]$F(0) = 32$[/tex], we have
[tex]$$
F(0) = a(0) + b = 32 \quad \Longrightarrow \quad b = 32.
$$[/tex]
Step 2. Find the slope [tex]$a$[/tex] using the second given point.
When [tex]$C = 100$[/tex], [tex]$F(100) = 212$[/tex]. Substitute into the linear function:
[tex]$$
F(100) = a(100) + 32 = 212.
$$[/tex]
Step 3. Solve for [tex]$a$[/tex].
Subtract 32 from both sides:
[tex]$$
100a = 212 - 32 = 180.
$$[/tex]
Divide both sides by 100:
[tex]$$
a = \frac{180}{100} = 1.8.
$$[/tex]
Thus, the linear function that expresses Fahrenheit in terms of Celsius is:
[tex]$$
F(C) = 1.8C + 32.
$$[/tex]
---
Part (a): Find the rate of change of Fahrenheit temperature for each unit change in Celsius.
The rate of change is the slope [tex]$a$[/tex], which is
[tex]$$
1.8 \text{ Fahrenheit degrees per Celsius degree}.
$$[/tex]
---
Part (b): Find and interpret [tex]$F(21)$[/tex].
Substitute [tex]$C = 21$[/tex] into the linear function:
[tex]$$
F(21) = 1.8(21) + 32.
$$[/tex]
Calculating,
[tex]$$
1.8 \times 21 = 37.8,
$$[/tex]
so
[tex]$$
F(21) = 37.8 + 32 = 69.8.
$$[/tex]
This means that when the Celsius temperature is [tex]$21^\circ\text{C}$[/tex], the Fahrenheit temperature is [tex]$69.8^\circ\text{F}$[/tex].
---
Part (c): Find [tex]$F(-30)$[/tex].
Substitute [tex]$C = -30$[/tex] into the linear function:
[tex]$$
F(-30) = 1.8(-30) + 32.
$$[/tex]
Calculating,
[tex]$$
1.8 \times (-30) = -54,
$$[/tex]
so
[tex]$$
F(-30) = -54 + 32 = -22.
$$[/tex]
Thus, when the Celsius temperature is [tex]$-30^\circ\text{C}$[/tex], the Fahrenheit temperature is [tex]$-22^\circ\text{F}$[/tex].
---
Summary of Answers:
1. The linear function is:
[tex]$$
F(C) = 1.8C + 32.
$$[/tex]
2. The rate of change is [tex]$1.8$[/tex] Fahrenheit degrees per Celsius degree.
3. [tex]$F(21) = 69.8$[/tex], meaning that [tex]$21^\circ\text{C}$[/tex] is equivalent to [tex]$69.8^\circ\text{F}$[/tex].
4. [tex]$F(-30) = -22$[/tex].