Question

Temperature Conversion The linear function \(F(C)=\frac{9}{5} C+32\) converts degrees Celsius to degrees Fahrenheit, and the linear function \(R(F)=F+459.67\) converts degrees Fahrenheit to degrees Rankine. Find a linear function that converts degrees Rankine to degrees Celsius.

Short answer

C(R) = \(\frac{5}{9} (R - 491.67)\)

Step-by-step solution

- Understand the Given Functions

The problem provides two conversion functions: 1. Celsius to Fahrenheit: \( F(C) = \frac{9}{5} C + 32 \) 2. Fahrenheit to Rankine: \( R(F) = F + 459.67 \)

- Combine the Given Functions

First, combine the two functions by substituting the first function into the second. So, if we have \( F(C) = \frac{9}{5} C + 32 \), since \( R(F) = F + 459.67 \), then \( R(C) = (\frac{9}{5} C + 32) + 459.67 \).

- Simplify the New Function

Combine like terms in the equation \( R(C) = \frac{9}{5} C + 32 + 459.67 \). Adding the constants: \( R(C) = \frac{9}{5} C + 491.67 \).

- Invert the Function to Solve for Celsius

To transform from Rankine to Celsius, we need to solve for \( C \) in terms of \( R \). Start with: \( R = \frac{9}{5} C + 491.67 \). First, subtract 491.67 from both sides: \( R - 491.67 = \frac{9}{5} C \) Next, multiply both sides by \( \frac{5}{9} \): \( C = \frac{5}{9} (R - 491.67) \).

- Write the Final Linear Function

The final linear function that converts degrees Rankine (\( R \)) to degrees Celsius (\( C \)) is: \( C(R) = \frac{5}{9} (R - 491.67) \).

Key concepts

Linear Functions

A linear function describes a relationship where each x value corresponds to a y value in a straight line when plotted on a graph.
The general form is often written as: \( y = mx + b \)
Here:

• \( m \) is the slope, showing how steep the line is.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

Both the Celsius to Fahrenheit and Fahrenheit to Rankine conversions are linear functions. Here, the slope and y-intercept help us change temperatures seamlessly.

Celsius to Fahrenheit Conversion

This conversion helps us understand temperatures in the Fahrenheit scale based on Celsius inputs. The linear function for this is: \( F(C) = \frac{9}{5} C + 32 \).
Let's break this down:

• \( \frac{9}{5} \) is the slope, meaning Fahrenheit increases by \( \frac{9}{5} \) units for every 1-degree increase in Celsius.
• 32 is the y-intercept, indicating that 0°C is equal to 32°F.

This is useful for various scientific, engineering, and everyday applications where you deal with different temperature scales.

Fahrenheit to Rankine Conversion

The Fahrenheit to Rankine conversion is even simpler: \( R(F) = F + 459.67 \).
This formula indicates that Rankine is just an adjusted Fahrenheit scale.

• The slope here is 1, meaning Fahrenheit and Rankine increase equally.
• 459.67 is the conversion constant, added to translate from Fahrenheit to Rankine.

This helps in fields where absolute temperature scales are needed, like thermodynamics.

Composite Functions

Composite functions involve applying one function to the results of another function.
If you have two functions, \(f(x)\) and \(g(x)\), the composite function is written as \((g \, \circ \, f)(x)\), or \(g(f(x))\).
In this problem, we started with two functions: converting Celsius to Fahrenheit \((F(C))\) and converting Fahrenheit to Rankine \((R(F))\). By substituting \( F(C) \) into \( R(F) \), we formed a composite function \(R(C)\). This helps us directly convert Celsius to Rankine in one step: \(R(C) = \frac{9}{5} C + 491.67 \).

Function Inversion

Function inversion helps us find an inverse function which reverses the effect of the original function.
If you have \( y = f(x)\), the inverse function \( f^{-1}(y)\) satisfies \( x = f^{-1}(y)\).
For Rankine to Celsius conversion, we inverted the function \(R = \frac{9}{5} C + 491.67\).

• First, we isolated \( C \) by subtracting 491.67 from both sides: \( R - 491.67 = \frac{9}{5} C \).
• Then, we multiplied by \( \frac{5}{9} \) to solve for \( C \): \(C = \frac{5}{9} (R - 491.67)\).

This gives us a convenient formula for converting Rankine directly to Celsius.