To determine if the temperatures in degrees Celsius and their equivalents in degrees Fahrenheit vary directly, Corrine can use the following steps:
### Step 1: Understanding Ratios
First, let's compare the temperatures by calculating their ratios. The direct variation implies that the ratio between Fahrenheit and Celsius temperatures should remain constant.
For the given Celsius (C) and Fahrenheit (F) pairs:
- (-10, 14)
- (5, 41)
- (10, 50)
- (20, 68)
### Step 2: Calculate the Fahrenheit temperatures using the conversion formula
Using the formula for converting Celsius to Fahrenheit:
[tex]\[ F = \left( \frac{9}{5} \times C \right) + 32 \][/tex]
### Step 3: Calculate the actual Fahrenheit temperatures and compare the ratios
Let's convert the given Celsius values to Fahrenheit using the formula and compare each with the corresponding given Fahrenheit value to calculate the ratio.
- For [tex]\( C = -10 \)[/tex]:
[tex]\[ F = \left( \frac{9}{5} \times -10 \right) + 32 = -18 + 32 = 14 \][/tex]
[tex]\[ \text{Ratio} = \frac{14}{14} = 1.0 \][/tex]
- For [tex]\( C = 5 \)[/tex]:
[tex]\[ F = \left( \frac{9}{5} \times 5 \right) + 32 = 9 + 32 = 41 \][/tex]
[tex]\[ \text{Ratio} = \frac{41}{41} = 1.0 \][/tex]
- For [tex]\( C = 10 \)[/tex]:
[tex]\[ F = \left( \frac{9}{5} \times 10 \right) + 32 = 18 + 32 = 50 \][/tex]
[tex]\[ \text{Ratio} = \frac{50}{50} = 1.0 \][/tex]
- For [tex]\( C = 20 \)[/tex]:
[tex]\[ F = \left( \frac{9}{5} \times 20 \right) + 32 = 36 + 32 = 68 \][/tex]
[tex]\[ \text{Ratio} = \frac{68}{68} = 1.0 \][/tex]
### Step 4: Verify if ratios are equivalent
The ratios for all pairs are found to be:
[tex]\[ [1.0, 1.0, 1.0, 1.0] \][/tex]
Since all ratios are equal, we can determine that the relationship forms a direct variation.
### Conclusion
Corrine can confirm that the temperatures vary directly because the ratios between the Fahrenheit and Celsius temperatures are consistent and equivalent for all given temperature pairs.
