Answer :
To find the inverse formula for converting temperatures from degrees Fahrenheit to degrees Celsius, we start with the given Celsius to Fahrenheit conversion formula:
[tex]\[ F(C) = \frac{9}{5}C + 32. \][/tex]
Our goal is to solve for [tex]\( C \)[/tex] in terms of [tex]\( F \)[/tex], effectively finding the inverse function [tex]\( C(F) \)[/tex].
Here are the steps:
1. Start with the original formula:
[tex]\[ F = \frac{9}{5}C + 32. \][/tex]
2. Isolate the term involving [tex]\( C \)[/tex]:
To do this, subtract 32 from both sides of the equation:
[tex]\[ F - 32 = \frac{9}{5}C. \][/tex]
3. Solve for [tex]\( C \)[/tex]:
To isolate [tex]\( C \)[/tex], multiply both sides by the reciprocal of [tex]\(\frac{9}{5}\)[/tex], which is [tex]\(\frac{5}{9}\)[/tex]:
[tex]\[ C = \frac{5}{9}(F - 32). \][/tex]
Now, we have the formula to convert a temperature from degrees Fahrenheit to degrees Celsius:
[tex]\[ C(F) = \frac{5}{9}F - \frac{5}{9} \times 32. \][/tex]
Calculating [tex]\( \frac{5}{9} \times 32 \)[/tex] gives us approximately 17.7778.
Therefore, the formula simplifies to:
[tex]\[ C(F) = 0.55555F - 17.7778. \][/tex]
This gives you the temperature in degrees Celsius when you start with a temperature in degrees Fahrenheit.
[tex]\[ F(C) = \frac{9}{5}C + 32. \][/tex]
Our goal is to solve for [tex]\( C \)[/tex] in terms of [tex]\( F \)[/tex], effectively finding the inverse function [tex]\( C(F) \)[/tex].
Here are the steps:
1. Start with the original formula:
[tex]\[ F = \frac{9}{5}C + 32. \][/tex]
2. Isolate the term involving [tex]\( C \)[/tex]:
To do this, subtract 32 from both sides of the equation:
[tex]\[ F - 32 = \frac{9}{5}C. \][/tex]
3. Solve for [tex]\( C \)[/tex]:
To isolate [tex]\( C \)[/tex], multiply both sides by the reciprocal of [tex]\(\frac{9}{5}\)[/tex], which is [tex]\(\frac{5}{9}\)[/tex]:
[tex]\[ C = \frac{5}{9}(F - 32). \][/tex]
Now, we have the formula to convert a temperature from degrees Fahrenheit to degrees Celsius:
[tex]\[ C(F) = \frac{5}{9}F - \frac{5}{9} \times 32. \][/tex]
Calculating [tex]\( \frac{5}{9} \times 32 \)[/tex] gives us approximately 17.7778.
Therefore, the formula simplifies to:
[tex]\[ C(F) = 0.55555F - 17.7778. \][/tex]
This gives you the temperature in degrees Celsius when you start with a temperature in degrees Fahrenheit.
