Problem 6 - Basic Technical Mathematics with Calculus

Question

Represent the data graphically. The temperatures felt by the body as a result of the wind-chill factor for an outside temperature of \(20^{\circ} \mathrm{F}\) (as determined by the National Weather Service) are given in the following table: $$\begin{array}{l|c|c|c|c|c|c|c|c} \text {Wind speed }(\mathrm{mi} / \mathrm{h}) & 5 & 10 & 15 & 20 & 25 & 30 & 35 & 40 \\ \hline \text {Temp. felt }\left(^{\circ} \mathrm{F}\right) & 13 & 9 & 6 & 4 & 3 & 1 & 0 & -1 \end{array}$$

Step-by-Step Solution

Answer

Plot data points on a graph with wind speed on the x-axis and perceived temperature on the y-axis, then connect them.

Step 1: Understand the Data

We have a set of temperatures as perceived by the body (the wind-chill temperatures) corresponding to different wind speeds. The given table shows that when the actual temperature is \(20^{\circ} \mathrm{F}\), the perceived temperature changes with wind speed.

Step 2: Gather Plot Requirements

To graph this data, identify the two variables: wind speed (\(\mathrm{mi/h}\)) and perceived temperature (\(\,^{\circ} \mathrm{F}\)). Wind speed will be plotted on the x-axis, and perceived temperature will be on the y-axis.

Step 3: Create the Graph

Using graph paper or software, set up the x-axis for wind speed ranging from 5 to 40 \(\mathrm{mi/h}\) and the y-axis for perceived temperature ranging from -5 to 15 \(\,^{\circ} \mathrm{F}\). Plot points for each pair of data from the table: (5,13), (10,9), (15,6), (20,4), (25,3), (30,1), (35,0), and (40,-1).

Step 4: Connect Data Points

Once all points are plotted on the graph, draw a smooth line connecting them. This will visually show how the perceived temperature changes as the wind speed increases.

Step 5: Label the Graph

Add the title 'Wind-Chill Temperature vs. Wind Speed' to the graph. Label the x-axis as 'Wind Speed (mi/h)' and the y-axis as 'Temperature Felt (\(\,^{\circ} \mathrm{F}\))'. Ensure that all data points are clearly visible and accurately plotted.

Key Concepts

Graph Plotting

Graph plotting is a key skill in understanding data insights visually.
In the context of our exercise, we are plotting a set of wind-chill temperatures against various wind speeds. For a successful plot, it is crucial to choose appropriate scales and ranges for the axes.

The x-axis represents the wind speed, which ranges from 5 to 40 mi/h. The y-axis portrays the temperature felt, ranging from -5 to 15 degrees Fahrenheit.
Each pair of numbers from the table provided corresponds to a point on the graph. For instance, the pair (5, 13) is plotted on the graph such that 5 is on the x-axis and 13 is on the y-axis. Similarly, each data point from the table should be plotted.

Once all points are plotted, these can be connected with a smooth line, representing how changes in wind speed affect the perceived temperature. This line makes it easy to understand the trend visually.
Label your axes clearly and add an informative title like "Wind-Chill Temperature vs. Wind Speed" to make your graph more user-friendly. A well-labeled graph is more intuitive to read and analyze.

Wind-Chill Effect

The wind-chill effect occurs when wind causes the air temperature to feel colder than it actually is. Wind removes the warm layer of air that surrounds your body, effectively lowering your body temperature when outside.

For example, at an external temperature of 20°F with increasing wind speeds, the temperature felt decreases significantly.

• At 5 mi/h, it feels like 13°F.
• As the wind speed increases to 40 mi/h, it may feel as low as -1°F.

This demonstrates the wind-chill's impact. The faster the wind, the cooler you feel due to this cooling effect.

Understanding the wind-chill effect is essential for safety during cold weather conditions, as it helps people prepare adequately by knowing how cold it will feel when exposed to the elements rather than just the actual air temperature.

Mathematical Modelling

Mathematical modelling involves representing real-world phenomena using mathematical concepts and equations. In this exercise, the relationship between wind speed and perceived temperature is modelled mathematically.

• Step one is to gather reliable data, such as the wind-chill temperatures for various wind speeds.
• Next, we visualize this data using a graph to discover patterns.

The plotted data points and the smooth line through them represent a simple mathematical model, illustrating the wind speed and perceived cold temperature correlation.

Mathematical models like this one can be used to predict future outcomes under similar circumstances. By understanding the model, you can estimate how wind speed at different levels might impact perceived cold, allowing for better preparedness under harsh weather conditions.