An Application of a Concept in Meteorology
Suppose an x-y axis is superimposed on the map with its origin at Des Moines, Iowa. Also suppose a cold front is passing through the United States at noon having a shape roughly like a parabola with its vertex at Des Moines, Iowa and a vertical stretch/shrink factor of if both longitudinal and latitudinal distances are measured in hundreds of miles. See the figure above.
Step 1: Answers the following questions.
a) What type of function should be used to model the storm front? How did you know?
b) Give both the general and standard forms of the type of function you gave as your answer to question one.
c) For the storm front shown in the figure above, is the leading coefficient positive or negative?
How did you know?
d) Keeping the x-y axis superimposed on the map with its origin at Des Moines, Iowa, what is the equation for the parabola that would represent the cold front?
Step 2: State the answer to the following question and also explain the process by which you arrived at your answer.
If the cold front is moving south at 40 mph for 4 hours and retains its present shape, what would be the equation of its graph at that time?
(Remember: both longitudinal and latitudinal distances are measured in hundreds of miles.)
Step 3: State the answer to the question below and also explain the process by which you arrived at your answer.
Suppose that by midnight the vertex of the front has moved 250 miles south and 210 miles east of Des Moines, maintaining the same shape. Write the equation for the parabola that would represent the cold front at midnight. (Remember: both longitudinal and latitudinal distances are measured in hundreds of miles.)
Step 4: Using computer technology (www.desmos.com or other mathematics graphing website) attach a graph of the parabola at midnight on the x-y axis keeping Des Moines at the origin and label the location of Columbus, Memphis, and Louisville on the graph. (Points may be added by hand to the computer generated graph).
Columbus, Ohio is 550 miles east and 80 miles south of Des Moines;
Memphis, Tennessee is 190 miles east and 430 miles south of Des Moines; and
Louisville, Kentucky is 420 miles east and 230 miles south of Des Moines.
Step 5: Make conclusions.
- For each of the above cities, state whether, by midnight, the city has already been affected by the front, is just about to be affected by the front, or has not yet been affected by the front.
- How are your conclusions supported by the graph you drew in Step 4.
c) Support your conclusions using algebraic methods.
