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].
