# Vectors, graphs, and counting

Project 4A

1. Explain why vectors QR and RQ are not equivalent.

2. Explain in your own words when the elimination method for solving a system of equations is preferable to the substitution method.

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.

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?

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.)

3b. If you buy one ticket, what is your probability of winning? (Note: You are picking six different numbers.)

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

Project 5A

1. 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.

2. Search Internet sources to find and discuss at least two practical uses of probability today. Be sure to cite your sources.

#### Solution Preview

1. Explain why vectors QR and RQ are not equivalent:

Vectors are basically specified by thier lenghts and directions. If two vecotrs have the same lenghts and directions, then they are equivalent. Now, QR is in the direction connecting Q to R, but RQ is in the opposite direction i.e. R to Q. Therefore, their directions are not the same, although they have the same lenghts.

2. Explain in your own words when the elimination method for solving a system of equations is preferable to the substitution method.

Well, basically the elimination method is a better method especially if we are dealing with a huge system like 10 equations in 10 unknowns. Moreover, in some particular cases when a system has no solutions or infinitely many solutions the elimination method can reveal that quickly while the substitution mehtod needs more work. For small systems like 2 equations in 2 unknowns these two methods are more or less equal.

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.

The equation of an ellipse is as ...

#### Solution Summary

There are several types of probabilities here. One uses lottery tickets to explore counting techniques, one discusses graphs of parabolas and hyperbolas, as well as applications of parabolas, and one discusses equivalent vectors.