Problem 65 - A Graphical Approach to College Algebra

Question

The table shows equivalent temperatures in degrees Celsius and degrees Fahrenheit. $$\begin{array}{c|c|c|c|c|c}\circ \mathrm{F} & -40 & 32 & 59 & 95 & 212 \\\\\hline^{\circ} \mathrm{C} & -40 & 0 & 15 & 35 & 100\end{array}$$ (a) Plot the data by having the \(x\) -axis correspond to Fahrenheit temperature and the \(y\) -axis to Celsius temperature. What type of relation exists between the data? (b) Find a function \(C\) that uses the Fahrenheit temperature \(x\) to calculate the corresponding Celsius temperature. Interpret the slope. (c) What is a temperature of \(83^{\circ} \mathrm{F}\) in degrees Celsius?

Step-by-Step Solution

Answer

The relation is linear; \(C = \frac{5}{9}x - 17.78\); 83°F is 28.33°C.

Step 1: Understanding the Graph

To analyze the relationship between Fahrenheit and Celsius temperatures, we need to plot the given data. On the x-axis, we map the temperatures in Fahrenheit: -40, 32, 59, 95, and 212. On the y-axis, we map the corresponding Celsius temperatures: -40, 0, 15, 35, and 100. Upon plotting these points, you will notice that they lie on a straight line, indicating a linear relationship between Fahrenheit and Celsius temperatures.

Step 2: Finding the Linear Function

The relationship between Fahrenheit (F) and Celsius (C) can be expressed as a linear equation, usually in the form \( C = mx + b \), where \(x\) represents temperature in Fahrenheit. To find \(m\), the slope, we use the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Take two points such as (32, 0) and (212, 100): \[ m = \frac{100 - 0}{212 - 32} = \frac{100}{180} = \frac{5}{9} \]. The y-intercept \(b\) can be determined from the point (32, 0); hence, \(b = -\frac{5}{9} \times 32 = -17.78\). So the function is \( C(x) = \frac{5}{9}x - 17.78 \).

Step 3: Interpreting the Slope

The slope \(\frac{5}{9}\) in the equation \( C = \frac{5}{9}x - 17.78 \) represents the rate of change of Celsius temperature with respect to Fahrenheit. This means that for every 1-degree increase in Fahrenheit, the Celsius temperature increases by \(\frac{5}{9}\) degrees.

Step 4: Using the Function to Find Celsius Temperature

To find the Celsius temperature equivalent of 83°F, we substitute 83 into the function: \(C(83) = \frac{5}{9}(83) - 17.78 \). Calculating the result, \[ C(83) = \frac{5 \times 83}{9} - 17.78 = 46.11 - 17.78 = 28.33 \]. Thus, 83°F is equivalent to approximately 28.33°C.

Key Concepts

Celsius to Fahrenheit

Temperature conversion between Celsius and Fahrenheit is a common need, especially in science classes or when traveling internationally. The two temperature scales measure the same thing—temperature—but they use different units.
To convert a temperature from degrees Fahrenheit (°F) to degrees Celsius (°C), we use the linear conversion formula:

• \( C = \frac{5}{9} (F - 32) \)
• \( F = \frac{9}{5}C + 32 \)

These formulas show us how the temperature in one scale relates to the other. Simply put, the linear relationship signifies that if you have a Fahrenheit temperature, you can easily use these formulas to find out what that temperature is in Celsius, and vice versa.
This conversion is essential for understanding how temperature impacts things in different parts of the world, and knowing both scales can be incredibly useful in both academic and practical life scenarios.

Linear Equations

Linear equations are a foundational concept in mathematics that describe a straight-line relationship between two variables.
In a linear equation of the form \( y = mx + b \), \(m\) represents the slope, and \(b\) is the y-intercept. The slope \(m\) tells us how steep the line is, while the y-intercept \(b\) shows where the line crosses the y-axis.
When applied to temperature conversion tasks, we model the relationship between Fahrenheit and Celsius with a linear equation. This is because, when plotted, the points representing Fahrenheit and Celsius temperatures form a straight line.

• For instance, using the points (32, 0) and (212, 100), which are known reference points for water's freezing and boiling points, allows us to determine the slope and y-intercept.
• This forms the basis of our linear equation for conversion: \( C(x) = \frac{5}{9}x - 17.78 \).

Understanding linear equations helps us grasp how changing one element (e.g., temperature in Fahrenheit) affects another (temperature in Celsius). It’s a crucial concept that transcends temperature conversion, applicable in various mathematical problems and real-world scenarios.

Slope Interpretation

Interpreting the slope in a linear equation provides insight into how two variables relate to each other. The slope illustrates the rate of change between these variables.
For the temperature conversion equation \( C = \frac{5}{9}F - 17.78 \), the slope \( \frac{5}{9} \) reveals how much the Celsius temperature changes for each 1-degree increase in Fahrenheit.

• This means for every degree the Fahrenheit temperature rises, the Celsius temperature increases by \( \frac{5}{9} \) of a degree.

Understanding the slope is crucial in scenarios beyond temperature conversion. Whether you are analyzing financial data or studying natural phenomena, interpreting slopes helps you understand the dynamics between variables.
By conceptualizing slope as a rate of change, students can better predict outcomes and understand the relationships they are studying.