Question

Temperature Conversion The function \(C(F)=\frac{5}{9}(F-32)\) converts a temperature in degrees Fahrenheit, \(F\). to a temperature in degrees Celsius, \(C\). The function \(K(C)=C+273,\) converts a temperature in degrees Celsius to a temperature in kelvins, \(K\). (a) Find a function that converts a temperature in degrees Fahrenheit to a temperature in kelvins. (b) Determine 80 degrees Fahrenheit in kelvins.

Short answer

The function is \(K(F) = \frac{5}{9}(F-32) + 273\). For 80°F, the temperature in kelvins is 299.67 K.

Step-by-step solution

- Understand the Given Functions

The given functions are: 1. The function that converts Fahrenheit to Celsius: \[C(F) = \frac{5}{9}(F-32)\] 2. The function that converts Celsius to Kelvin: \[K(C) = C + 273\]

- Express the Function to Convert Fahrenheit to Kelvin

To convert Fahrenheit to Kelvin, combine the two functions. First, convert Fahrenheit to Celsius using \(C(F)\), then convert the result to Kelvin using \(K(C)\). The composite function is: \[K(F) = K(C(F))\] Substitute \(C(F)\) into \(K(C)\): \[K(F) = \left(\frac{5}{9}(F-32)\right) + 273\]

- Simplify the Function

Simplify the expression to get the final function: \[K(F) = \frac{5}{9}(F-32) + 273\]

- Determine the Kelvin Temperature for 80°F

Use the function \(K(F)\) to find the temperature in kelvins when \(F = 80\): \[K(80) = \frac{5}{9}(80 - 32) + 273\] Calculate the value inside the parentheses: \[80 - 32 = 48\] Now, calculate the fraction: \[\frac{5}{9} \times 48 = 26.67\] Finally, add 273 to the result: \[26.67 + 273 = 299.67\]

Key concepts

Fahrenheit to Celsius conversion

Temperature conversion can be confusing at first, but it gets easier once you understand the formulas. Let's start with the basics: converting Fahrenheit to Celsius.

To convert a temperature from Fahrenheit to Celsius, we use the formula: \( C(F) = \frac{5}{9}(F - 32) \).

Here's a breakdown of the formula:

• Subtract 32 from the Fahrenheit temperature \( (F - 32) \).
• Multiply the result by \( \frac{5}{9} \).

For example, to convert 80°F to Celsius: \[ C(80) = \frac{5}{9}(80 - 32) = \frac{5}{9} \times 48 = 26.67 \]So, 80°F is approximately 26.67°C.

Remember, the key part of this conversion is knowing you need to subtract 32 first, then multiply by \( \frac{5}{9} \).

Celsius to Kelvin conversion

Converting Celsius to Kelvin is straightforward. You just need to add 273 to the Celsius temperature:

The formula for this conversion is: \( K(C) = C + 273 \).

This is because Kelvin is an absolute temperature scale, where 0 Kelvin represents absolute zero, the theoretical point where atomic movement ceases.

For example, if you have a temperature of 26.67°C (which we previously found from 80°F), you convert it to Kelvin like this: \[ K(26.67) = 26.67 + 273 = 299.67 \]So, 26.67°C is equal to 299.67 K.

Notice how simple this conversion is? Just a single addition, but it gives meaningful information when dealing with scientific measurements and calculations.

Composite functions

Composite functions combine two or more functions into one. They are useful for chaining multiple conversions, like in our temperature example.

For converting Fahrenheit to Kelvin, we first convert Fahrenheit to Celsius and then Celsius to Kelvin. This is an example of a composite function.

The steps are:

• Convert Fahrenheit to Celsius using \( C(F) = \frac{5}{9}(F - 32) \).
• Take the result and convert it to Kelvin using \( K(C) = C + 273 \).

To combine these steps, we can write the composite function as: \[ K(F) = K(C(F)) = \left( \frac{5}{9}(F - 32) \right) + 273 \]
For our example of converting 80°F to Kelvin:

• First, find Celsius: \( C(80) = \frac{5}{9} \times (80 - 32) = 26.67 \).
• Then, convert Celsius to Kelvin: \( K(26.67) = 26.67 + 273 = 299.67 \).

Putting it all together, we have: \[ K(80) = \left( \frac{5}{9}(80 - 32) \right) + 273 = 299.67 \]Understanding composite functions makes handling such calculations much easier.