Let's address the problem step by step:
### Part (a):
Question: What Celsius temperatures correspond to Fahrenheit temperatures of [tex]\(212^{\circ}\)[/tex] and [tex]\(98.6^{\circ}\)[/tex]? What is special about these two temperatures? Are there any other special temperatures?
Solution:
We are given the formula to convert Fahrenheit to Celsius:
[tex]\[ C = \frac{5}{9} (F - 32) \][/tex]
Let's plug in the given Fahrenheit temperatures one by one.
1. For [tex]\(212^{\circ}F\)[/tex]:
[tex]\[ C = \frac{5}{9} (212 - 32) \][/tex]
[tex]\[ C = \frac{5}{9} \times 180 \][/tex]
[tex]\[ C = 100 \][/tex]
Hence, [tex]\(212^{\circ}F\)[/tex] converts to [tex]\(100^{\circ}C\)[/tex].
2. For [tex]\(98.6^{\circ}F\)[/tex]:
[tex]\[ C = \frac{5}{9} (98.6 - 32) \][/tex]
[tex]\[ C = \frac{5}{9} \times 66.6 \][/tex]
[tex]\[ C = 37 \][/tex]
Hence, [tex]\(98.6^{\circ}F\)[/tex] converts to [tex]\(37^{\circ}C\)[/tex].
Special Temperatures:
- [tex]\(212^{\circ}F\)[/tex] corresponds to [tex]\(100^{\circ}C\)[/tex], which is the boiling point of water at standard atmospheric pressure.
- [tex]\(98.6^{\circ}F\)[/tex] corresponds to [tex]\(37^{\circ}C\)[/tex], which is considered the average human body temperature.
These are indeed special temperatures due to their significance in everyday life. Other special temperatures could include the freezing point of water, which is [tex]\(32^{\circ}F\)[/tex] and corresponds to [tex]\(0^{\circ}C\)[/tex].
### Part (b):
Question: Solve the equation above for [tex]\(F\)[/tex] to obtain a rule for converting Celsius temperatures to Fahrenheit temperatures.
Solution:
To derive the formula for converting Celsius to Fahrenheit, we start with the given conversion formula:
[tex]\[ C = \frac{5}{9} (F - 32) \][/tex]
We need to solve this equation for [tex]\(F\)[/tex]. To do this, follow these steps:
1. Multiply both sides by [tex]\(\frac{9}{5}\)[/tex] to isolate [tex]\(F - 32\)[/tex] on the right side:
[tex]\[ \frac{9}{5}C = F - 32 \][/tex]
2. Add [tex]\(32\)[/tex] to both sides to solve for [tex]\(F\)[/tex]:
[tex]\[ F = \frac{9}{5}C + 32 \][/tex]
Therefore, the formula for converting Celsius to Fahrenheit is:
[tex]\[ F = 1.8C + 32 \][/tex]
In conclusion:
- Celsius temperatures corresponding to [tex]\(212^{\circ}F\)[/tex] and [tex]\(98.6^{\circ}F\)[/tex] are [tex]\(100^{\circ}C\)[/tex] and [tex]\(37^{\circ}C\)[/tex], respectively. These temperatures are special because they mark the boiling point of water and average human body temperature.
- The rule for converting Celsius to Fahrenheit temperatures is [tex]\( F = 1.8C + 32 \)[/tex].
