To determine the function that models the relationship between Fahrenheit (F) and Celsius (C), we start by noting the two given points: [tex]\( 68^\circ F \)[/tex] is equivalent to [tex]\( 20^\circ C \)[/tex], and [tex]\( 86^\circ F \)[/tex] is equivalent to [tex]\( 30^\circ C \)[/tex].
These points indicate a linear relationship between Fahrenheit and Celsius. Therefore, we can use the formula for a straight line, which is given by [tex]\( F = mC + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
1. Calculate the slope (m):
The slope [tex]\( m \)[/tex] is determined by the change in Fahrenheit divided by the change in Celsius.
[tex]\[
m = \frac{F_2 - F_1}{C_2 - C_1}
\][/tex]
Substituting the given values:
[tex]\[
m = \frac{86 - 68}{30 - 20} = \frac{18}{10} = 1.8
\][/tex]
2. Calculate the y-intercept (b):
Using the point [tex]\( (C_1, F_1) = (20, 68) \)[/tex] and the slope [tex]\( m = 1.8 \)[/tex], we can use the equation [tex]\( F = mC + b \)[/tex] and solve for [tex]\( b \)[/tex]:
[tex]\[
68 = (1.8 \times 20) + b
\][/tex]
Simplifying inside the parentheses:
[tex]\[
68 = 36 + b
\][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[
b = 68 - 36 = 32
\][/tex]
Thus, the equation relating Fahrenheit to Celsius is:
[tex]\[
F = 1.8C + 32
\][/tex]
In the choices given, we see this matches with:
[tex]\[
F = \frac{9}{5}C + 32
\][/tex]
since [tex]\(1.8\)[/tex] is equivalent to [tex]\(\frac{9}{5}\)[/tex].
Therefore, the correct function that models the relationship between Fahrenheit and Celsius is:
[tex]\[
F = \frac{9}{5} C + 32
\][/tex]
