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.
