College

We appreciate your visit to When the Celsius temperature is tex 30 circ tex the corresponding Fahrenheit temperature is tex 86 circ tex When the Celsius temperature is tex 110. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!

When the Celsius temperature is [tex]30^{\circ}[/tex], the corresponding Fahrenheit temperature is [tex]86^{\circ}[/tex]. When the Celsius temperature is [tex]110^{\circ}[/tex], the corresponding Fahrenheit temperature is [tex]230^{\circ}[/tex]. Let [tex]C[/tex] represent the Celsius temperature and [tex]F[/tex] the Fahrenheit temperature.

a. Express [tex]F[/tex] as an exact linear function of [tex]C[/tex].
b. Solve the equation in part a for [tex]C[/tex], thus expressing [tex]C[/tex] as a function of [tex]F[/tex].
c. For what temperature is [tex]F = C[/tex]?

a. Express [tex]F[/tex] as an exact linear function of [tex]C[/tex].
[tex] F = \square [/tex]
(Simplify your answer. Do not include the degree symbol in your answer.)

Answer :

Let's solve the problem step-by-step, using the information we have about temperatures in Celsius and Fahrenheit.

### Part a: Express [tex]\( F \)[/tex] as a linear function of [tex]\( C \)[/tex]

The problem gives us two points for temperature conversion:
- When Celsius ([tex]\( C \)[/tex]) is 30, Fahrenheit ([tex]\( F \)[/tex]) is 86.
- When Celsius ([tex]\( C \)[/tex]) is 110, Fahrenheit ([tex]\( F \)[/tex]) is 230.

We can represent these points as [tex]\((30, 86)\)[/tex] and [tex]\((110, 230)\)[/tex].

Step 1: Find the slope [tex]\( m \)[/tex] of the line

The slope [tex]\( m \)[/tex] of a line through two points [tex]\((C_1, F_1)\)[/tex] and [tex]\((C_2, F_2)\)[/tex] is given by:

[tex]\[
m = \frac{F_2 - F_1}{C_2 - C_1}
\][/tex]

Substitute the given points:

[tex]\[
m = \frac{230 - 86}{110 - 30} = \frac{144}{80} = 1.8
\][/tex]

Step 2: Find the y-intercept [tex]\( b \)[/tex]

Using the slope-intercept form [tex]\( F = mC + b \)[/tex], substitute one of the points, say [tex]\((30, 86)\)[/tex], to find [tex]\( b \)[/tex]:

[tex]\[
86 = 1.8 \times 30 + b
\][/tex]

[tex]\[
86 = 54 + b
\][/tex]

[tex]\[
b = 86 - 54 = 32
\][/tex]

Thus, the linear function is:

[tex]\[
F = 1.8C + 32
\][/tex]

### Part b: Solve for [tex]\( C \)[/tex] to express [tex]\( C \)[/tex] as a function of [tex]\( F \)[/tex]

We have [tex]\( F = 1.8C + 32 \)[/tex]. To express [tex]\( C \)[/tex] as a function of [tex]\( F \)[/tex], solve for [tex]\( C \)[/tex]:

1. Subtract 32 from both sides:

[tex]\[
F - 32 = 1.8C
\][/tex]

2. Divide both sides by 1.8:

[tex]\[
C = \frac{F - 32}{1.8}
\][/tex]

This can be simplified to:

[tex]\[
C = 0.5556F - 17.7778
\][/tex]

### Part c: Find the temperature where [tex]\( F = C \)[/tex]

Set the two expressions for [tex]\( F \)[/tex] and [tex]\( C \)[/tex] equal to each other:

[tex]\[
1.8C + 32 = C
\][/tex]

Rearrange terms:

[tex]\[
1.8C - C = -32
\][/tex]

[tex]\[
0.8C = -32
\][/tex]

Divide by 0.8:

[tex]\[
C = -40
\][/tex]

Therefore, the temperature at which [tex]\( F = C \)[/tex] is [tex]\(-40\)[/tex] degrees.

Thanks for taking the time to read When the Celsius temperature is tex 30 circ tex the corresponding Fahrenheit temperature is tex 86 circ tex When the Celsius temperature is tex 110. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada