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The function [tex]\( C(F) = \frac{5}{9}(F - 32) \)[/tex] is used to convert temperature from Fahrenheit [tex]\((F)\)[/tex] to Celsius [tex]\((C)\)[/tex]. The function [tex]\( K(C) = C + 273.15 \)[/tex] is used to convert temperature from Celsius [tex]\((C)\)[/tex] to the Kelvin [tex]\((K)\)[/tex] scale. Write a function to convert [tex]\( 77^{\circ} F \)[/tex] to the Kelvin scale.

To express temperature in Kelvin as a function of Fahrenheit, compose the functions as [tex]\( K(C(F)) \)[/tex].

Now, derive the function described above and use it to express [tex]\( 77^{\circ} F \)[/tex] as [tex]\(\square \, K\)[/tex].

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Sagot :

To solve the problem, let's proceed step-by-step:

1. Understanding the Functions:
- We have the function [tex]\( C(F) = \frac{5}{9}(F - 32) \)[/tex] which converts a temperature from Fahrenheit to Celsius.
- We also have the function [tex]\( K(C) = C + 273.15 \)[/tex] which converts a temperature from Celsius to Kelvin.

2. Composing the Functions:
- To convert directly from Fahrenheit to Kelvin, we need to substitute the Celsius function into the Kelvin function.

Therefore:
- [tex]\( K(F) = K(C(F)) \)[/tex]
- Substituting [tex]\( C(F) \)[/tex] into [tex]\( K \)[/tex]:
[tex]\[ K(F) = K\left(\frac{5}{9}(F - 32)\right) \][/tex]
- This gives us:
[tex]\[ K(F) = \frac{5}{9}(F - 32) + 273.15 \][/tex]

3. Calculating the Temperature in Kelvin:
- Now, we substitute [tex]\( F = 77 \)[/tex] into our composed function [tex]\( K(F) \)[/tex]:

[tex]\[ K(77) = \frac{5}{9}(77 - 32) + 273.15 \][/tex]
- Evaluating this expression, we obtain:
[tex]\[ K(77) = 25.0 + 273.15 = 298.15 \][/tex]

In summary:
1. The composed function to convert Fahrenheit directly to Kelvin is:
[tex]\[ \boxed{\frac{5}{9}(F - 32) + 273.15} \][/tex]
2. Using this function, [tex]\( 77^\circ F \)[/tex] is converted to:
[tex]\[ \boxed{298.15 \text{ K}} \][/tex]