Problem 65 - A Graphical Approach to Precalculus with Limits

Question

The table shows equivalent temperatures in degrees Celsius and degrees Fahrenheit. $$\begin{array}{|c|c|c|c|c|c|} \hline^{\circ} \mathbf{F} & -40 & 32 & 59 & 95 & 212 \\ \hline^{\circ} \mathrm{C} & -40 & 0 & 15 & 35 & 100 \end{array}$$ (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) Convert a temperature of \(86^{\circ} \mathrm{F}\) to degrees Celsius.

Step-by-Step Solution

Answer

The data shows a linear relationship; 86°F converts to 30°C using the function \(C(x) = \frac{5}{9}x - \frac{160}{9}\).

Step 1: Identify Data Points for Plotting

From the table, we have the data points \((-40, -40), (32, 0), (59, 15), (95, 35), (212, 100)\). These points should be plotted on a graph with the \(x\)-axis representing Fahrenheit and the \(y\)-axis representing Celsius.

Step 2: Plot Data Points on Graph

Plot the data points on a Cartesian plane using the pairs: \((-40, -40), (32, 0), (59, 15), (95, 35), (212, 100)\). Connect the points to visualize the relationship. The plot should show a linear relationship.

Step 3: Determine Type of Relationship

Observe the plotted points. Notice they lie on or very close to a straight line indicating a linear relationship between Fahrenheit and Celsius temperatures.

Step 4: Derive the Linear Equation

Use the two-point formula for a line, \(y = mx + b\), where \(m\) is slope and \(b\) is the y-intercept. Use points \((32, 0)\) and \((212, 100)\) to calculate the slope \(m\): \[m = \frac{100 - 0}{212 - 32} = \frac{100}{180} = \frac{5}{9}\] Using point \((32, 0)\), substitute in line equation for \(b\):\[0 = \frac{5}{9}(32) + b\]Solving, \(b = -\frac{160}{9}\). So, the line equation: \(C(x) = \frac{5}{9}x - \frac{160}{9}\).

Step 5: Interpret the Slope

The slope \(\frac{5}{9}\) indicates that for each 1 degree increase in Fahrenheit, Celsius increases by about \(\frac{5}{9}\) degrees.

Step 6: Use the Equation to Convert 86°F to °C

Substitute \(x = 86\) into\(C(x) = \frac{5}{9}x - \frac{160}{9}\). Solve for \(C\): \[C(86) = \frac{5}{9}(86) - \frac{160}{9} = \frac{430}{9} - \frac{160}{9} = \frac{270}{9} = 30\]. Thus, \(86\degree F\) converts to \(30\degree C\).

Key Concepts

Linear Relationship

When examining the relationship between Fahrenheit and Celsius temperatures, it quickly becomes apparent that there is a linear relationship between these two measurements. This means that as one temperature increases or decreases, the other does so in a consistent manner. You can visualize this by plotting the given temperature data on a graph.

To do this, use the Fahrenheit temperature on the x-axis and the Celsius temperature on the y-axis. Upon plotting the data points, you'll notice they form a straight line or lie very closely along a straight path. This straight alignment is a hallmark of a linear relationship, indicating that the change between Fahrenheit and Celsius is uniform and predictable. Linear relationships are straightforward, making them easy to analyze and perfect for understanding temperature conversions.

Slope Interpretation

The slope of the line in a linear relationship holds significant meaning. In the context of temperature conversion between Fahrenheit and Celsius, the slope represents the rate of change between these two units. Calculating the slope involves selecting two data points and applying the slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

For Fahrenheit and Celsius, the slope is calculated to be \frac{5}{9}\. What this tells us is for every 1-degree increase in Fahrenheit, the Celsius temperature increases by \frac{5}{9}\ degrees. Understanding this slope helps simplify temperature conversions, making it easier to switch between the two.

The consistent increment that the slope introduces is key to predicting temperatures. This calculable change is central to interpreting trends effortlessly in data sets with linear relationships.

Cartesian Graphing

When plotting data for temperature conversion, we use a Cartesian graphing system. This system uses two axes—the x-axis for the Fahrenheit temperatures and the y-axis for the Celsius temperatures.

This method of graphing allows for a clear visual representation of the relationship between the two temperature scales. By graphically displaying the data points, we can easily interpret patterns. Cartesian graphing is beneficial for visual learners who find it easier to understand concepts when they are displayed visually.

Once the data is plotted, connecting the points with a line helps to further emphasize the linear relationship. The Cartesian plane not only aids in plotting but also in deriving the equation of the line using the plotted points; this equation is crucial for converting temperatures efficiently.

Data Plotting

Data plotting involves taking your raw data, in this case, temperature pairs, and placing them on a graph to look for patterns or relationships. Plotting the temperatures from the table involves translating each pair of Fahrenheit and Celsius values into points on the graph.

Start by marking the x-axis with Fahrenheit values and the y-axis with Celsius values. Then, identify each temperature pair as a point (e.g., (-40, -40), (32, 0)) and plot these on the graph.

• A consistent and evenly-spaced axis helps in keeping the data legible.
• Connecting the points gives a line showing the linear relationship.
• This visual orderly display makes it easier to interpret and use the plotted data for further calculations, such as finding the equation of the line.

Understanding how to plot data accurately is an invaluable skill for analyzing the relationship between different data sets.