1.
In Washington DC, there is a large grassy area south of the White House known as the Ellipse. It
is actually an ellipse with major axis of length 1048 ft. and minor axis of length 898 ft. Assuming
that a coordinate system is superimposed on the area in such a way that the center is at the origin
and the major and minor axes are on the x- and y- axes of the coordinate system, respectively, find
an equation of the ellipse.
By hypothesis, we know that a coordinate system is superimposed on the area in
such a way that the center is at the origin and the major and minor axes are on the xand y- axes of the coordinate system, respectively. We know major axis of length 1048
ft. and minor axis of length 898 ft. So, we have the equation as follows.
x2
50242
2
y
 449
2 1
2. A family borrows $120,000. The loan is to be repaid in 13 years at 12% interest, compounded
annually. How much will be repaid at the end of 13 years?
By a formula, we know that the total be repaid is
120000 * (1  12%)13  120000 * 4.3635  $535620
3a. Twice a week, the state of Michigan runs a 6 out of 49 number lotto that pays at least $2 million.
You purchase a ticket for $1 and pick any 6 numbers from 1 to 49. If your numbers match those that
the state draws, you win. How many different 6-number combinations are there for the drawing? If you
buy one ticket, what is your probability of winning (6 out of 6)? (Note: You are picking six different
numbers.)
Since you pick 6 numbers and each number can be 1,2,…,49. Hence, there are
49*49*49*49*49*49=1.38412872*10^10
different 6-number combinations.
If you buy a ticket and win, you must choose those 6 numbers. So there are 6!=720
different ways to get them. So the probability is
P=720/(1.38412872*10^10)
=5.202*10^(-8)
3b. If you buy one ticket, what is your probability of winning? (Note: You are picking six different
numbers.)
We have got the probability is
P=720/(1.38412872*10^10)
=5.202*10^(-8)
3c. If this were a Super Lotto with the scenario listed below, how many different six number
combinations are there for the drawing?
• The first five numbers can be any number 1-49 (Note: You are picking five different numbers.)
• The sixth number can be any number, 1-49, including one of the numbers you picked in the first set of
five
The first five numbers can be any number 1-49 (Note: You are picking five different
numbers.)
The sixth number can be any number, 1-49, including one of the numbers you picked
in the first set of five
Since the first five numbers can be any number 1-49 (Note: You are picking five
different numbers.) , we have
M=49*48*47*46*45/5!=1906884
different five numbers.
Then the sixth number can be any number, 1-49, including one of the numbers you
picked in the first set of five, so there are 49 possible ways to choose this number. So,
we get
49*M=93437316
different six number combinations for the drawing
4. How does the graph of a parabola differ from the graph of one branch of a hyperbola. Identify some
real world applications of parabolas and hyperbolas.
It seems there is no difference between the graph of a parabola and the graph of one branch of a
hyperbola. But some times the graph of a parabola corresponds to a quadratic function such as y=x^2, its
graph is symmetric with y-axis. But a hyperbola usually corresponds to x^2/a^2-y^2/b^2=1, its graph is
symmetric with x-axis. So, we can only identify its graph by its equations.
In the real life, the orbits of planets are parabolas and radio antennas are parabolas as
well.
Dulles Airport, designed by Eero Saarinen, is in the shape of a hyperbolic
paraboloid. The hyperbolic paraboloid is a three-dimensional curve that is a hyperbola in
one cross-section, and a parabola in another cross section. See picture below.
5. Search the Cybrary or other Internet sources to find and discuss at least two practical uses of
probability today. Be sure to cite your sources.
Example 1 http://www.financewise.com/public/edit/energy/weather00/wthr00-forecast.htm
We use probability theory to forecast the weather, for example, if the data shows today is raining with
probability 0.8 and with probability 0.10 cloudy and with probability 0.10 sunny.
Example 2, Canada's
two main lotteries work (Lotto 6/49 and Super7 Lottery), the
same as your question 3, we can compute the probability and determine if you will
win after you buy a ticket.