Problem 34 - Calculus

Question

What function C(F) converts Fahrenheit to Celsius (or Centigrade), and what is its slope?

Step-by-Step Solution

Answer

The slope is \( \frac{5}{9} \).

Step 1: Understanding the Relationship

The relationship between the Fahrenheit and Celsius scales is linear. Given the known conversion formula: \[ C = \frac{5}{9}(F - 32) \] We need to express \( C \) as a function \( C(F) \) in terms of \( F \).

Step 2: Identifying the Function

From the conversion formula, the function \( C(F) \) that describes the relationship between Celsius and Fahrenheit is: \[ C(F) = \frac{5}{9}(F - 32) \] This can be rewritten in the form \( C(F) = \frac{5}{9}F - \frac{160}{9} \), but for this exercise, the initial form will suffice.

Step 3: Determining the Slope

The slope of a linear equation \( y = mx + b \) is \( m \). In the function \( C(F) = \frac{5}{9}(F-32) \), this is equivalent to \( C(F) = \frac{5}{9}F - \frac{160}{9} \). Thus, the slope \( m \) is \( \frac{5}{9} \).

Key Concepts

Linear Functions

A linear function is one of the simplest types of mathematical functions. It is expressed in the form of \( y = mx + b \), where \( m \) represents the slope and \( b \) represents the y-intercept. These functions create straight lines when plotted on a graph, which is why they are referred to as "linear."
Linear functions have key properties:

• Constant rate of change: The slope \( m \) defines how fast \( y \) changes with respect to \( x \).
• Straight-line graph: When graphed, a linear function will always produce a straight line.
• Two variables: Typically involves two variables, an independent variable \( x \) and a dependent variable \( y \).

Understanding linear functions helps in grasping many real-world applications, such as temperature conversion, as they model relationships where change occurs at a constant rate. In the case of converting Fahrenheit to Celsius, the relationship is linear, which we explore through the temperature conversion formula.

Slope Calculation

The slope is a crucial component of a linear equation, defining the steepness and direction of the line. It is denoted by \( m \) in the function \( y = mx + b \). We calculate the slope using the formula:
\[ m = \frac{\Delta y}{\Delta x} \]
which means the change in \( y \) over the change in \( x \).
In any linear function:

• If \( m > 0 \), the line ascends as it moves from left to right, indicating a positive relationship.
• If \( m < 0 \), the line descends, showing a negative relationship.
• If \( m = 0 \), the line is horizontal, meaning there is no change in \( y \) as \( x \) changes.

For our temperature conversion function \( C(F) = \frac{5}{9}(F - 32) \), the slope \( m \) is \( \frac{5}{9} \). This slope reflects the rate at which Celsius temperature changes as Fahrenheit changes.

Temperature Conversion Formula

Temperature conversion between Fahrenheit and Celsius is a classic example of applying a linear function. The formula
\[ C = \frac{5}{9}(F - 32) \]
converts Fahrenheit to Celsius by first subtracting 32 from the Fahrenheit value and then multiplying by \( \frac{5}{9} \). This formula arises from aligning the two temperature scales at specific points:

• 0°C aligns with 32°F, which is the freezing point of water.
• 100°C aligns with 212°F, the boiling point of water.

These reference points help set up the linear function that matches each Fahrenheit temperature with its Celsius equivalent.
Understanding this formula involves recognizing the linear nature of the temperature relationship, where the Celsius temperature can be seen as a function \( C(F) \). This provides a practical example of linear functions and slope calculation, making temperature conversion an excellent learning tool for these mathematical concepts.