Problem 72 - Algebra and Trigonometry

Question

Daily Temperature Change Temperature readings \(T\) (in F) were recorded every 2 hours from midnight to noon in Atlanta, Georgia, on March \(18,1996 .\) The time \(t\) was measured in hours from midnight. Sketch a rough graph of \(T\) as a function of \(t\) \(\begin{array}{|c|c|}\hline t & {T} \\ \hline 0 & {58} \\ {2} & {57} \\ {4} & {53} \\ {6} & {50} \\ {8} & {51} \\ {10} & {57} \\ {12} & {61} \\\ \hline\end{array}\)

Step-by-Step Solution

Answer

Plot the points from the data and connect them to visualize the temperature changes over time.

Step 1: Understand the Data

The table provides temperature readings at different times, measured in hours from midnight to noon. The times are 0, 2, 4, 6, 8, 10, and 12 hours, with corresponding temperatures of 58°F, 57°F, 53°F, 50°F, 51°F, 57°F, and 61°F respectively.

Step 2: Set Up the Axes

Draw a horizontal axis (x-axis) and a vertical axis (y-axis). Label the horizontal axis as time, \( t \) (in hours), and the vertical axis as temperature, \( T \) (in degrees Fahrenheit). Set the scale for time from 0 to 12 hours and for temperature from 50°F to 65°F.

Step 3: Plot the Data Points

For each pair \( (t, T) \), plot the corresponding point on the graph. Start with the point at \( (0, 58) \), then plot \( (2, 57) \), \( (4, 53) \), \( (6, 50) \), \( (8, 51) \), \( (10, 57) \), and \( (12, 61) \). Ensure the points accurately represent the temperature at each given time.

Step 4: Connect the Points

Draw a smooth line or curve connecting each plotted point in sequence from \( t = 0 \) to \( t = 12 \). This line represents the change in temperature over time. Note how the temperature decreases from midnight to early morning and then increases as it approaches noon.

Step 5: Analyze the Graph

Observe the overall shape of the graph. The temperature starts high, dips to a minimum at around \( t = 6 \), then rises again. This provides a visual representation of daily temperature changes in Atlanta for that specific period.

Key Concepts

Plotting Data

When you are given a set of data, like temperature readings at specific times, your first goal is to visualize this data. This process is known as plotting. By transforming tabular data into a graphical representation, you make it easier to observe trends and patterns. To start, you'll need to set up two axes:

• The x-axis represents the independent variable, which in this case is time (measured in hours from midnight).
• The y-axis represents the dependent variable, temperature, which changes over time.

Once the axes are prepared, you will directly plot each data point given in the dataset. For instance, on this graph, the coordinates (0, 58) reflect 58°F at 0 hours. Each point corresponds to a specific time and its associated temperature. Plotting these correctly ensures the accuracy of your graph.

Temperature Change

Understanding temperature change involves watching how temperature shifts over a period. In our example, observing data from Atlanta, you notice a common pattern in daily temperature changes. Here's a breakdown of the observed changes:

• The temperature begins at 58°F at midnight, hinting at a warmer evening end.
• It then drops to its lowest point of 50°F at 6 hours.
• After reaching this minimum, temperatures start to increase as the day progresses, eventually reaching 61°F by noon.

This sequence illustrates a typical diurnal cycle where the temperature dips during the early hours and rises as the sun ascends. Recognizing these patterns can help in making educated guesses about future temperature trends.

Graph Analysis

Analyzing a graph provides insights into the data's story. Once your temperature data is plotted and points are connected, you're ready to dive into analysis. Start by examining the graph's shape:

• Notice the initial decline in temperature from midnight to 6 am. This is depicted by a downward slope on the graph.
• The ascent from 6 am to noon creates an upward slope, clearly showing an increase in temperature.

By observing these movements, you can infer the temperature's least and most stable periods. The trough at 6 am signifies the lowest temperature point, while the peak at noon reveals the highest. Understanding these elements helps in painting the bigger picture of daily climatic conditions and can assist in weather forecasting and planning.