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.
What type of function should be used to model the storm front? How did you know?
A parabolic function should be used during modeling a storm font due to the shape of the font and the manner in which it moves along an axis.
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Give both the general and standard forms of the type of function you gave as your answer to question one.
f (x) = a(x – h) 2 + k - standard form, (h,k) is the vertex
F( x) = ax 2 + bx + c, a ≠ , - quadratic form of the equation
c) For the storm front shown in the figure above, is the leading coefficient positive or negative?
How did you know?
Positive because it is found on the positive side of the Y-axis
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?
The equation will be: f (x) = a(x-0)(x-0), when y-intercept is 0
0= a(0-0)(0-0)
Solving for a = 0
The equation for the parabola will be f(x) = 0
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.)
Vertex will be at (0,-4), and Y-intercept at (0,-4)
The vertex form of the function will be f(x) = a(x(x+4) – 4
Using the Y-Intercept (0,-2)
-4 = a(0+4) -4 to find a =0, The quadratic function becomes
F(x) = -4
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.)
Vertex Coordinates (210,-250)
Y= a(x-210) 2 -250, = a(x 2 + 420x- 44350)
Since h=-b/2a, therefore, a=-b/2h = 420/420 =1
The quadratic equation becomes f(x) = x 2 + 420x- 44350
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: Des Moines, Columbus, Memphis and Louisville are the four American cities which were referenced in the above step. For each of these cities, write a paragraph which includes highlights regarding the history and / or culture of that city. Choose a single city statistic (for example: average temperature, population, number of restaurants, number of professional sports teams, etc.) and write a fifth paragraph comparing this statistic for the four cities.
Memphis is known for its culture of river boards, innovative African-American music, and the existence of a culinary of barbeque. Its population is approximately 245,836 people and a population density of 726.4 per square kilometer.
Columbus is the capital of Ohio State and has a population of approximately 850,000 people. The main metropolitan areas in the city include: Marion, Chillicothe, and Mount Vernon. The economy of the city is diverse and is characterized by education, insurance, and agriculture. It is also known for its role in sports activities such as sailing ( Brill, 2014) .
Louisville is the 30 th most populous city in the United States and is the largest in Kentucky. It has a humid subtropical climate and experiences the four seasons of: string, summer, autumn, and winter. The temperature of the city is hot and humid at approximately 32 o C. There are periods when the average temperature of the city if 32 o C. Religion of the people is a combination of Christians, Judaism, Islam and Hinduism. The most common economic activity in Memphis is shipping and cargo handling. A number of annual events are held in the county such as the Kentucky Derby that features fireworks.
When Louisville is compared with other cities such as Columbus, the former has a significant number of shipping activities compared with the latter. However, the temperature of Louisville has been less stable during most of the periods compared with other states such as Memphis and Columbus that have low temperature ranges ( Capaldi, 2014) .
Step 6: Make conclusions.
In the case of Memphis, Tennessee, it has already been affected by the font because it is closer to the path of the font compared with other cities. This means that when the font occurs, there is a high chance of its being affected compared with Louisville or Columbus. This is as illustrated in figure 2 which shows that Columbus and Louisville are further away from the path of the font compared with Memphis.
The algebraic equation representing the fonts passing through Memphis is:
f(x) = 190 2 + 420*190- 44350 = 71550
The equation that represents the font passing through Columbus is:
f(x) = 550 2 + 420*550- 44350 = 489,150
In the case of Louisville it is further away from the path of the font and is yet to be affected by it. There are also low chances of it being affected by the font.
The equation for the font passing through Louisville is:
f(x) = 420 2 + 420*420- 44350 = 308,450
As illustrated in the equations, the value of f(x) of Memphis is smaller than that of Louisville and Columbus, which shows that it is the closest to the fonts.
References
Brill, K. F. (2014). Revisiting an Old Concept: The Gradient Wind*. Monthly Weather Review , 142 (4), 1460-1471.
Capaldi, M. (2014). Non-traditional methods of teaching abstract algebra. PRIMUS , 24 (1), 12-24.
