Corrine wrote temperatures in degrees Celsius and the equivalent temperatures in degrees Fahrenheit.

\begin{tabular}{|c|c|c|c|c|}
\hline \multicolumn{5}{|c|}{ Equivalent Temperatures } \\
\hline Celsius & -10 & 5 & 10 & 20 \\
\hline Fahrenheit & 14 & 41 & 50 & 68 \\
\hline
\end{tabular}

Which explains how Corrine could determine if the temperatures vary directly?

A. Compare the equivalent temperatures as a ratio; if the ratios are equivalent, then the temperatures vary directly.
B. Draw a graph of the temperatures; if the points form a straight line, then the temperatures vary directly.
C. Write an equation that converts one temperature to another; if the temperatures satisfy the equation, then they vary directly.
D. Look for a pattern in the table; if the pattern is consistent, then the temperatures vary directly.



Answer :

Let's carefully analyze the problem of determining if the given Celsius and Fahrenheit temperatures vary directly.

### Step-by-Step Solution:

1. Analyze the given temperatures:

We have the following pairs of equivalent temperatures:
- Celsius: -10, 5, 10, 20
- Fahrenheit: 14, 41, 50, 68

2. Determine whether the temperatures vary directly:

In order to check if the temperatures vary directly, we need to see if there is a consistent relationship between the Celsius and Fahrenheit temperatures. The best way to determine this is through the standard equation for converting Celsius to Fahrenheit:

[tex]\[ F = \frac{9}{5} C + 32 \][/tex]

Where [tex]\( C \)[/tex] is the temperature in Celsius and [tex]\( F \)[/tex] is the temperature in Fahrenheit.

3. Verify the relationship for each pair of temperatures:

Let's check each pair to see if the conversion follows the given equation:

- For [tex]\( C = -10 \)[/tex]:
[tex]\[ F = \frac{9}{5} (-10) + 32 = -18 + 32 = 14 \][/tex]
[tex]\[ Correspondence: (-10, 14) \][/tex]

- For [tex]\( C = 5 \)[/tex]:
[tex]\[ F = \frac{9}{5} 5 + 32 = 9 + 32 = 41 \][/tex]
[tex]\[ Correspondence: (5, 41) \][/tex]

- For [tex]\( C = 10 \)[/tex]:
[tex]\[ F = \frac{9}{5} 10 + 32 = 18 + 32 = 50 \][/tex]
[tex]\[ Correspondence: (10, 50) \][/tex]

- For [tex]\( C = 20 \)[/tex]:
[tex]\[ F = \frac{9}{5} 20 + 32 = 36 + 32 = 68 \][/tex]
[tex]\[ Correspondence: (20, 68) \][/tex]

4. Draw a conclusion:

After verifying the relationship for each pair, we see that all these pairs satisfy the conversion equation [tex]\( F = \frac{9}{5} C + 32 \)[/tex].

5. Summarize:

- The ratios between Celsius and Fahrenheit temperatures are consistent and follow the standard conversion equation.
- Since each pair of temperatures fits the conversion formula perfectly, we can conclude that the temperatures vary directly.

Thus, Corrine can determine that the temperatures vary directly by checking that each pair of equivalent temperatures satisfies the standard conversion formula [tex]\( F = \frac{9}{5} C + 32 \)[/tex].