Answer :
To determine whether a temperature in degrees Celsius is always proportional to its equivalent temperature in degrees Fahrenheit, we need to analyze the conversion formula.
The formula to convert Celsius to Fahrenheit is:
[tex]\[ F = C \times \frac{9}{5} + 32 \][/tex]
Where:
- [tex]\( F \)[/tex] is the temperature in Fahrenheit.
- [tex]\( C \)[/tex] is the temperature in Celsius.
### Understanding Proportionality
Two quantities are said to be proportional if there is a constant [tex]\( k \)[/tex] such that one quantity is always [tex]\( k \)[/tex] times the other. In mathematical terms, if [tex]\( y \)[/tex] is proportional to [tex]\( x \)[/tex], then:
[tex]\[ y = k \times x \][/tex]
### Applying this to our Formula
Let's rewrite the formula for converting Celsius to Fahrenheit in terms of proportionality:
[tex]\[ F = C \times \frac{9}{5} + 32 \][/tex]
If we compare this with the proportional relationship equation [tex]\( y = k \times x \)[/tex], we'll notice an additional term, [tex]\( +32 \)[/tex], which means that the relation is not simply [tex]\( F = k \times C \)[/tex].
### Analysis
1. Proportional Relationship Test: For [tex]\( F \)[/tex] to be proportional to [tex]\( C \)[/tex]:
- The formula should be in the form [tex]\( F = k \times C \)[/tex].
2. Current Formula:
- Here, [tex]\( F \)[/tex] is not just [tex]\( k \times C \)[/tex]. There is an additional constant term [tex]\( +32 \)[/tex].
This constant term [tex]\( +32 \)[/tex] indicates a shift in the linear relationship, implying that the temperature conversion involves not just a multiplication by a constant factor, but also an addition of a constant value.
### Conclusion
Since there is an extra term [tex]\( +32 \)[/tex] in the conversion formula from Celsius to Fahrenheit, the relationship is not proportional. Therefore, a temperature in degrees Celsius is not always proportional to its equivalent temperature in degrees Fahrenheit.
The formula to convert Celsius to Fahrenheit is:
[tex]\[ F = C \times \frac{9}{5} + 32 \][/tex]
Where:
- [tex]\( F \)[/tex] is the temperature in Fahrenheit.
- [tex]\( C \)[/tex] is the temperature in Celsius.
### Understanding Proportionality
Two quantities are said to be proportional if there is a constant [tex]\( k \)[/tex] such that one quantity is always [tex]\( k \)[/tex] times the other. In mathematical terms, if [tex]\( y \)[/tex] is proportional to [tex]\( x \)[/tex], then:
[tex]\[ y = k \times x \][/tex]
### Applying this to our Formula
Let's rewrite the formula for converting Celsius to Fahrenheit in terms of proportionality:
[tex]\[ F = C \times \frac{9}{5} + 32 \][/tex]
If we compare this with the proportional relationship equation [tex]\( y = k \times x \)[/tex], we'll notice an additional term, [tex]\( +32 \)[/tex], which means that the relation is not simply [tex]\( F = k \times C \)[/tex].
### Analysis
1. Proportional Relationship Test: For [tex]\( F \)[/tex] to be proportional to [tex]\( C \)[/tex]:
- The formula should be in the form [tex]\( F = k \times C \)[/tex].
2. Current Formula:
- Here, [tex]\( F \)[/tex] is not just [tex]\( k \times C \)[/tex]. There is an additional constant term [tex]\( +32 \)[/tex].
This constant term [tex]\( +32 \)[/tex] indicates a shift in the linear relationship, implying that the temperature conversion involves not just a multiplication by a constant factor, but also an addition of a constant value.
### Conclusion
Since there is an extra term [tex]\( +32 \)[/tex] in the conversion formula from Celsius to Fahrenheit, the relationship is not proportional. Therefore, a temperature in degrees Celsius is not always proportional to its equivalent temperature in degrees Fahrenheit.
