Problem 94 - Algebra and Trigonometry Real Mathematics, Real People

Question

Meterology Recall that water freezes at \(0^{\circ} \mathrm{C}\left(32^{\circ} \mathrm{F}\right)\) and boils at \(100^{\circ} \mathrm{C}\left(212^{\circ} \mathrm{F}\right)\). (a) Find an equation of the line that shows the relationship between the temperature in degrees Celsius \(C\) and degrees Fahrenheit \(F\). (b) Use the result of part (a) to complete the table. $$\begin{array}{|c|c|c|c|c|c|c|} \hline C & & -10^{\circ} & 10^{\circ} & & & 177^{\circ} \\ \hline F & 0^{\circ} & & & 68^{\circ} & 90^{\circ} & \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The equation that shows the relationship between the temperature in degrees Celsius \(C\) and degrees Fahrenheit \(F\) is \(F = 1.8C + 32\). The completed table would read as follows: [-10°C, 14°F], [10°C, 50°F], [177°C, 350.6°F].

Step 1: Find the Linear Equation

The boiling and freezing points of water give two points on the graph: (0°C, 32°F) and (100°C, 212°F). Our goal is to find a linear equation that describes the relationship between Celsius and Fahrenheit temperatures. To find the slope and the y-intercept for the line, we can utilize the slope-intercept form of the linear equations \(y = mx + b\), where \(y\) represents Fahrenheit \(F\), \(m\) is the slope of the line, \(x\) represents Celsius \(C\), and \(b\) is the y-intercept. We begin by finding the slope \(m\) using two points - point 1 with coordinates (0, 32) and point 2 with coordinates (100, 212), and the formula \(m = (y2 - y1)/(x2 - x1)\). Substituting these points into the equation we get \(m = (212-32)/(100-0) = 180/100 = 1.8\). Since at 0°C the Fahrenheit temperature is 32, we know that the y-intercept \(b\) is 32. So, the linear equation relating \(C\) and \(F\) is \(F = 1.8C + 32\).

Step 2: Complete the Table

Now that we have our equation \(F = 1.8C + 32\), we can use it to fill in the missing Fahrenheit temperatures for the respective Celsius temperatures in the table. We can plug in -10°C, 10°C and 177°C into this equation and calculate the corresponding Fahrenheit temperatures. Hence we obtain, for -10°C to Fahrenheit: \(F = 1.8*(-10) + 32 = 14\), for 10°C to Fahrenheit: \(F = 1.8*10 + 32 = 50\), and for 177°C to Fahrenheit: \(F = 1.8* 177 + 32 = 350.6\).

Key Concepts

Linear Equations

A linear equation represents a straight line on a graph. This equation shows a constant rate of change and is commonly written as \( y = mx + b \). Here, \( y \) and \( x \) are variables, \( m \) represents the slope of the line, and \( b \) is the y-intercept, where the line crosses the y-axis. By using two different temperature points—like the freezing and boiling points of water—we can calculate the relationship between Celsius and Fahrenheit temperatures. This creates a straightforward method to understand how changes in one variable affect the other.

Slope-Intercept Form

The slope-intercept form \( y = mx + b \) is an easy and popular way to express linear equations. It provides us with two key pieces of information: the slope \( m \) and the y-intercept \( b \).

• The **slope** \( m \) indicates how steep the line is. In context, the slope describes how much the Fahrenheit temperature changes per one degree Celsius.
• The **y-intercept** \( b \) is the point where the line crosses the y-axis, representing the Fahrenheit equivalent at 0 degrees Celsius, which is 32°F.

The slope-intercept form helps us quickly understand the relationship between two variables and predict values easily.

Celsius to Fahrenheit

To convert Celsius to Fahrenheit, we use the linear equation \( F = 1.8C + 32 \). Each Celsius degree translates into a 1.8 degree change in Fahrenheit. Adding 32 aligns the scale so that at 0°C, it's equivalent to 32°F.

Let's break it down:

• Start by multiplying the Celsius temperature (\( C \)) by 1.8.
• Add 32 to this result to get the Fahrenheit temperature (\( F \)).

For example, if you have 10°C, plug it into the equation: \( F = 1.8 \times 10 + 32 = 50°F \). This formula provides a straightforward way of converting Celsius to Fahrenheit, which is especially useful in science and daily weather forecasts.

Fahrenheit to Celsius

Converting Fahrenheit to Celsius reverses the process, using the equation \( C = \frac{F - 32}{1.8} \). This allows us to determine how many Celsius degrees correspond to a given Fahrenheit temperature.

Here is how it works:

• Subtract 32 from the Fahrenheit temperature to adjust the starting point.
• Divide the result by 1.8 to convert it to Celsius degrees.

For example, if you have 68°F, the conversion calculation would be \( C = \frac{68 - 32}{1.8} = 20°C \). Understanding this conversion is crucial for scientific data interpretation and understanding foreign weather reports.