Exploration of Hurricane Data:
AN ALGEBRA 2 PROJECT

Data Source: http://www.nhc.noaa.gov/2002epac.shtml
Project adapted from similar project by Mr. G Engel
http://www.mr-engel-math.com/

Question: Is there a relationship between the air pressure in a tropical storm or hurricane and the wind speed of the storm?

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Project Requirements:

1. Using the data from hurricanes Douglas and Kenna on the Hurricane Data Tables page, convert the wind speed to miles per hour and organize a data table that will pair pressure (in mb) and wind speed (in mph). Include on each data table a pair for the mean pressure and mean wind speed .
   [Create separate tables for each hurricane.]
   
2. For each hurricane, plot the data from your pressure and wind speed table as a scatter plot (by hand). The data should follow a linear pattern. Fit a line of best fit to the data. Calculate the slope of the line and the equation of the line. Express your prediction model as a function of wind speed in terms of pressure, .
   
3. Use one of your mathematical models to determine the wind speed (in mph) for the following pressure levels: 925, 950, 975, 1000, 1025, and 1050. Which of these predictions are interpolations and which are extrapolations? Discuss the reasonableness of the predictions.
   
4. Research: Using appropriate research materials (all resources must be cited), write a one-page, typed summary on the Saffir-Simpson Hurricane Scale. Include in your summary the answer the following question: A Category Three Hurricane has winds 111 mph-130 mph, what range of air pressure would you expect for a hurricane of this force? Support your answer with appropriate mathematics (the "algebra" may be done by hand on separate paper).

[Style Requirements for your typed paper: double-spaced, standard margins, Font: Times New Roman, regular style, size 12]

[Click on the Image for a larger version |
Click here for more info on Hurricane Andrew, 1992]

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A Worked Example

Below is a table of data for a hurricane from 2002. From the raw data, it appears that there is a relationship between air pressure in a storm and the wind speed: the greater the pressure, the lower the speed. Can we find a mathematical formula to describe this pattern?

There are different types of “basic” graphs in math, with several examples shown below.  Until we see a scatter plot of the data, it is difficult to decide which pattern the data fits (if it fits any of the patterns).

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The data we are looking at are the pressure and wind speed.  In this project we are work­ing with “paired data,” pairing pressure and wind speed. Other examples of paired data are “time and distance,” “diameter and circumference,” “hours worked and earnings,” “number of vertices of a polygon and number of diagonals,” and so on. Our goal is to plot the data pairs like (x, y) pairs and to see if the data points follow a pattern that we can analyze with math skills. Thus, first we will reorganize the data into a more convenient table, taking out unnecessary information.

Next, let us convert the wind speed in knots to miles per hour. The difference is not very great, but "miles per hour" is more readily understandable for most people since knots is measure of speed on water (knots = nautical miles per hour). To do this, multiply each speed in knots by the conversion factor for miles per hour: 1.15. Present this new information in a chart that you will show alongside your graph. Also, we are going to see that the points fall in approximately a straight-line pattern. The line that will best fit the pattern will pass through a point that is the mean of all the pressure-values and the wind speed-values. Thus, add to the table a point for mean-pressure and mean-wind speed. This is denoted p-bar and W-bar. Here this point is calculated to be .

Pressure (mb)	Wind Speed (mph)
1004	34.5
1004	34.5
1004	40.25
997	57.5
984	80.5
970	103.5
974	97.75
1001	40.25

Sketch the Scatter Plot

You must first set up a pair of axes with appropriate scales. Often paired data involve an independent variable and a dependent variable. This occurs often in science experiments, such as time and the amount of a chemical absorbed by a medical patient. In a case like that, it is clear that the amount of the chemical absorbed "depends" on the time. Generally, with paired data elements the horizontal axis records the independent variable information and the vertical axis the dependent variable. Since we cannot control time that is often the independent variable. In our situation, does the air pressure control the wind speed or does the wind speed cause changes in pressure? This is difficult to answer so we will arbitrarily assign the horizontal axis to pressure and the vertical axis to wind speed. More simply stated, we let pressure be our x-values and wind speed the y-values.

Setting up the Axes with Appropriate Scales

Setting up the coordinate axes with appropriate scales is one of the most important skills that you must master. Looking at our "x-values," the pressure has a minimum of 970 and a maximum of 1004. Our horizontal axis must have this range of values. I recommend going a bit below and a bit above the minimum and maximum for "elbow room." Also, we are probably going to try to make predictions that go below and above our data, if we chop the graph off at the ends, we cannot make these predictions with the graph. How far above and below do we want to go? Many students mistakenly assume they must start every graph at 0, but this not true. In this case, because the pressure is from 970 and 1004, it makes little sense to extend the graph all the way back to zero.

 

Once the range of x- and y-values has been determined, the axes must be broken up with "tick" marks. On my graph the y-values will go from 0 to 120. I do not want to mark by units because the axis will be too crowded and will be difficult to read. Thus, I've decided to make a tick mark at every 10. Below you will see the graph I have decided on, using WinPlot. This graph is called a scatter plot. For your Hurricane Project, we are practicing graphing skills and you are required to do the graph "by hand" on graph paper. Be sure to appropriately label your axes. Notice that the mean value point is labeled to draw attention to it.

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Analyzing the Graph

The graph shows that high wind speeds occur when the pressure is low and that as pressure increases, the wind speed decreases. The data seems to follow a path that is almost a straight-line. One of the most popular ways to express the equation of a line is the slope-intercept form, and for this we need the slope of the line. Slope, m, is the ratio of the vertical change in data to the horizontal change.

We could arbitrarily select two points from the data to find the slope, but we want to find the line that best approximates the data. This is called the line of best fit. To do this we use the mean value point for one of the two points. Using a ruler and pivoting the ruler with the mean value point as the center, we approximate a line that follows the data pattern. The line is drawn and a second point is selected. The second point should be some distance away from the mean value point and is selected because it is either on the line we drew or is very close to the line. Here I have selected the point as the other point to find the slope.

To find the equation of the line, the best method is to use the point-slope form of a linear equation:

Making Predictions Based on a Mathematical Model

The equation of this line of best fit, called the mathematical model, can be used to make predictions about the wind speed of a tropical storm / hurricane if given the air pressure. First, it makes sense to change the notation and to write it in function style, with W = the speed of the wind in mph and p = the air pressure in millibars:

The graph shown here displays the data as a scatter plot along with the prediction model. It appears the model is a good fit to the data because it follows the linear path of the data points.

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Using the Prediction Model

The project requirement # 3 asks you to use a prediction model "to determine the wind speed (in mph) for the following pressure levels: 925, 950, 975, 1000, 1025, and 1050." This can be done by simply substituting these values for p into . A graphing calculator's table feature simplifies this process even further:

Reading this table tells us that according to our prediction equation, if the pressure is 925 mb, the wind speed is approximately 197.68 mph, if the pressure is 975 mb, the wind speed will be approximately 95.68 mph, and so on. The project instructions also ask you to determine which of these predictions are interpolations and which are extrapolations and to discuss the reasonableness of the predictions.

 

A prediction is an interpolation if it is made between the lowest and highest values of data used to create the prediction equation. Here, the lowest pressure from the actual hurricane data was 970 mb and the highest was 1004 mb. Any prediction made between these values is an interpolation. Thus, the predictions for pressure of 975 and 1000 are interpolations.

 

A prediction is an extrapolation if it is made below the lowest or above highest values of data used to create the prediction equation. Thus, the predictions for 925, 950, 1025, and 1050 mb are all extrapolations. Extrapolations are always less reliable than interpolations and the further the prediction is from the upper and lower boundaries, the less reliable the prediction will be. Here we see predictions of wind speeds that are negative for pressure values of 1025 and 1050. "Negative speed" is absurd, thus these predictions are not reasonable.

 

In requirement # 4 you are asked to predict the pressure if you know the wind speed is between 111 mph and 130 mph. This is accomplished by setting up the equation with the speed equal to the prediction equation and then solving for p.

Thus, the pressure level that will produce 111 mph winds is approximately 967.5 mb.

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For Additional Review on Linear Equations:

• Linear Equations Review Notes

Image Credits:

• hcane_andrew2: http://www.nhc.noaa.gov/HAW2/english/history.shtml#andrew
• hcane_andrewOverhead_sm: http://rsd.gsfc.nasa.gov/rsd/images/andrew.html

Math Connections

It is astounding how mathematical properties show up in places in the real-world that seem like they should have nothing in common. For example, is there anything in common between a hurricane on Earth and the structure of a galaxy millions of light year away? Yes, there is! The structure of the spiral shape of hurricanes is very similar to the spiral shape of many galaxies. Click here to learn more. This is just one more reason to continue to study math and science.

For more information on hurricanes, you might try: How Hurricanes Work.

For information on Hurricane Katrina, August 2005.

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	Matthew C. Whitney