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 :

To determine if the temperatures in degrees Celsius and their equivalent temperatures in degrees Fahrenheit vary directly, we can approach this problem with the steps outlined:

1. Compare the Equivalent Temperatures as Ratios:

We will calculate the ratio of each Fahrenheit value (F) to its corresponding Celsius value (C).

[tex]\[ \text{Ratios: } \frac{14}{-10}, \frac{41}{5}, \frac{50}{10}, \frac{68}{20} \][/tex]

This gives us the following ratios:

[tex]\[ \frac{14}{-10} = -1.4, \quad \frac{41}{5} = 8.2, \quad \frac{50}{10} = 5.0, \quad \frac{68}{20} = 3.4 \][/tex]

Since these ratios are not equivalent, the temperatures do not vary directly.

2. Draw a Graph of the Temperatures:

Plotting the given Celsius and Fahrenheit values on a graph:

- (-10, 14)
- (5, 41)
- (10, 50)
- (20, 68)

If we plot these points on a graph and observe that they do not form a straight line, it indicates that they do not vary directly.

3. Write an Equation that Converts One Temperature to Another:

There is a standard equation that converts Celsius to Fahrenheit:

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

We can verify this equation with the given temperatures.

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

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

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

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

Since all points satisfy the equation \( F = \frac{9}{5}C + 32 \), the temperatures do indeed vary according to this equation, but not directly as a simple ratio.

4. Look for a Pattern in the Table:

When examining the pattern:

[tex]\[ -10 \rightarrow 14, \quad 5 \rightarrow 41, \quad 10 \rightarrow 50, \quad 20 \rightarrow 68 \][/tex]

The increases do not follow a consistent ratio, indicating that they do not vary directly.

In conclusion, Corrine's temperatures do satisfy the equation [tex]\( F = \frac{9}{5}C + 32 \)[/tex], which confirms the relationship between Celsius and Fahrenheit. The temperatures do not vary directly as simple ratios, but they do follow a consistent linear pattern as defined by the standard conversion formula.