Problem 82 - Thinking Mathematically

Question

The relationship between Celsius temperature, \(C\), and Fahrenheit temperature, \(F\), can be described by a linear equation in the form \(F=m C+b\). The graph of this equation contains the point \((0,32)\) : Water freezes at \(0^{\circ} \mathrm{C}\) or at \(32^{\circ} \mathrm{F}\). The line also contains the point \((100,212)\) : Water boils at \(100^{\circ} \mathrm{C}\) or at \(212^{\circ} \mathrm{F}\). Write the linear equation expressing Fahrenheit temperature in terms of Celsius temperature.

Step-by-Step Solution

Answer

The linear equation which expresses the Fahrenheit temperature in terms of the Celsius temperature is F = 1.8C + 32.

Step 1: Compute the slope

Given two points on a line (x1, y1) and (x2, y2), the slope (m) of the line is calculated as (y2-y1)/(x2-x1). Here x represents the Celsius temperature and y represents the Fahrenheit temperature. So here, x1 = 0, y1 = 32 (for the freezing point) and x2 = 100, y2 = 212 (for the boiling point). So the slope is m = (212-32)/(100-0) = 180/100 = 1.8.

Step 2: Substitute the slope in the equation

Substitute m = 1.8 in the equation F = mC + b to get F = 1.8C + b.

Step 3: Compute the y-intercept

Now substitute in the equation the coordinates of any of the two points, for example, the freezing point of water (0, 32). This gives 32 = 1.8*0 + b. Solving this for b gives b = 32.

Step 4: Write the final equation

Substitute b = 32 in the equation F = 1.8C + b to get final equation as F = 1.8C + 32.

Key Concepts

Slope Calculation in Linear Equations

In understanding linear equations, the concept of slope calculation is fundamental. The slope represents the rate at which the y-value changes for every unit increase in the x-value.

To calculate the slope, you need two points that lie on the line. With coordinates \( (x_1, y_1) \) and \( (x_2, y_2) \), the formula for slope \( m \) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). In practical scenarios like temperature conversion, this shows us how much the Fahrenheit temperature \( (F) \) increases when the Celsius temperature \( (C) \) increases by one degree.

For example, from the freezing point \( (0^\circ C, 32^\circ F) \) to the boiling point \( (100^\circ C, 212^\circ F) \) of water, we find the slope as follows: \( m = \frac{(212 - 32)}{(100 - 0)} = \frac{180}{100} = 1.8 \). This means that for each degree increase in Celsius, the Fahrenheit temperature increases by 1.8 degrees. \

Understanding this calculation is crucial in various scientific and engineering fields where relationships between two quantities need to be quantified.

Temperature Conversion Using Linear Equations

Temperature conversion between Celsius and Fahrenheit is a practical application of linear equations. Based on the fixed points of water's freezing and boiling temperatures, we can derive a straightforward equation to convert temperatures from one scale to another.

This linear relationship is described by the formula \( F = mC + b \), where \( m \) is the slope and \( b \) is the y-intercept. Once you've determined the slope as outlined in the slope calculation, you’re half way through. Using the slope \( (m = 1.8) \) and the y-intercept \( (b) \) that will be identified through one of the known points, we can write the equation that directly converts Celsius to Fahrenheit.

For instance, knowing that at \( 0^\circ C \) the temperature in Fahrenheit is \( 32^\circ F \) gives us a point to determine \( b \) and hence the full conversion equation. This kind of linear model is not just useful for thermometry; similar equations are used everywhere from finance to physics.

Y-intercept Computation in Linear Equations

The y-intercept of a linear equation is the value of \( y \) when \( x = 0 \) – it's where the line crosses the y-axis. To compute the y-intercept, also denoted as \( b \) in the equation \( y = mx + b \) – you can use one of the points that the line passes through.

For temperature conversion, when we substitute the known freezing point of water \( (0, 32) \) into our linear equation \( F = 1.8C + b \), we get \( 32 = 1.8\times0 + b \). From this, it's clear that \( b = 32 \), implying that when \( C = 0 \) (meaning at 0 degrees Celsius), \( F \) is 32. This is logical as 32 degrees Fahrenheit is the freezing point of water.

In contexts beyond temperature conversion, finding the y-intercept is a vital step in graphing a linear equation or understanding the starting point of a relationship between variables. For students tackling algebra or coordinate geometry, mastering this step is essential for solving a wide array of problems.