To determine which function models the relationship between Fahrenheit (F) and Celsius (C), we need to use the given information and analyze each function to see which one correctly converts Celsius to Fahrenheit.
Given:
- [tex]\( 20^{\circ} C \)[/tex] is equivalent to [tex]\( 68^{\circ} F \)[/tex]
- [tex]\( 30^{\circ} C \)[/tex] is equivalent to [tex]\( 86^{\circ} F \)[/tex]
Let's evaluate each function step-by-step using the given temperature conversions:
1. Function: [tex]\( F = \frac{5}{9}C - \frac{512}{9} \)[/tex]
- For [tex]\( C = 20 \)[/tex]:
[tex]\[
F = \frac{5}{9} \times 20 - \frac{512}{9} = \frac{100}{9} - \frac{512}{9} = \frac{100 - 512}{9} = \frac{-412}{9} \approx -45.78
\][/tex]
- For [tex]\( C = 30 \)[/tex]:
[tex]\[
F = \frac{5}{9} \times 30 - \frac{512}{9} = \frac{150}{9} - \frac{512}{9} = \frac{150 - 512}{9} = \frac{-362}{9} \approx -40.22
\][/tex]
Neither of these results match the given Fahrenheit values, so this function is incorrect.
2. Function: [tex]\( F = \frac{9}{5}C - \frac{\pi+2}{6} \)[/tex]
- For [tex]\( C = 20 \)[/tex]:
[tex]\[
F = \frac{9}{5} \times 20 - \frac{\pi + 2}{6} = 36 - \frac{\pi + 2}{6}
\][/tex]
This doesn't yield a straightforward integer due to the term involving [tex]\(\pi\)[/tex], and calculation shows it doesn't match [tex]\(68^{\circ} F\)[/tex].
- For [tex]\( C = 30 \)[/tex]:
[tex]\[
F = \frac{9}{5} \times 30 - \frac{\pi + 2}{6} = 54 - \frac{\pi + 2}{6}
\][/tex]
Again, this calculation does not match [tex]\(86^{\circ} F\)[/tex].
3. Function: [tex]\( F = \frac{9}{5}C + 32 \)[/tex]
- For [tex]\( C = 20 \)[/tex]:
[tex]\[
F = \frac{9}{5} \times 20 + 32 = 36 + 32 = 68
\][/tex]
This value matches the given [tex]\( 68^{\circ} F \)[/tex].
- For [tex]\( C = 30 \)[/tex]:
[tex]\[
F = \frac{9}{5} \times 30 + 32 = 54 + 32 = 86
\][/tex]
This value matches the given [tex]\( 86^{\circ} F \)[/tex].
4. Function: [tex]\( F = \frac{9}{8}C - 32 \)[/tex]
- For [tex]\( C = 20 \)[/tex]:
[tex]\[
F = \frac{9}{8} \times 20 - 32 = 22.5 - 32 = -9.5
\][/tex]
- For [tex]\( C = 30 \)[/tex]:
[tex]\[
F = \frac{9}{8} \times 30 - 32 = 33.75 - 32 = 1.75
\][/tex]
Neither of these results match the given Fahrenheit values, so this function is incorrect.
Based on the analysis above, the function that correctly models the relationship between Fahrenheit and Celsius is:
[tex]\[ \boxed{F = \frac{9}{5}C + 32} \][/tex]
