Taylor+Carroll

**Taylor Carroll - April 15, 2011**

 * 1.** Rene Descartes contributed many items to mathematics, such as analytical geometry, the theory of vortices, using the first letters in the alphabet to represent known numbers, and using the last letters in the alphabet to represent the unknown numbers. He also invented the rule of signs, he helped develop Cartesian geometry, and he introduced the way we write powers today. He was a man that contributed many, many items to mathematics, but these examples are some of his most prevalent.


 * 2.** Descartes contributed many items to mathematics that we still use today. He invented analytical geometry and he was the first person to use letters in the alphabet to represent known quantities and unknown quantities. He also introduced the way we write powers today. Without his contributions we would not be able to calculate equations the way we do today. He helped pave the way for mathematics and helped the topic flourish. His contributions will live on in mathematics for the rest of time.


 * 3.** Maria Gaetana Agnesi was raised in a very woman oppressive time; Women were not permitted to a higher education and were told that women in higher learning was a mark of sin. In Italy (where Maria Gaetana Agnesi was born) women were encouraged to become educated and educated women were almost "worshiped" by men. Maria Gaetana Agnesi was no exception, she was extremely gifted in mathematics and was a very kind-hearted woman. She took care of sick and dying women in shelters up until she herself died. Maria Gaetana Agnesi also did not focus solely on mathematics, mathematics was a hobby for her and she never really wanted to become famous.


 * 4.** One concept that was interesting and useful to me was the distance formula.

This formula makes it very easy to find the distance between points and solve point distance equations. This formula was also very easy for me to understand and solve. I know I have seen "find the distance between each point" equations on the practice ACT tests. This formula makes it easy for you to solve these types of equations. For example:


 * Calculate the distance between (4,–3) and (–3,8). (This ACT practice problem was found on []) ||

To solve this equation you have to plug your numbers into the distance formula.

So, the distance between the points is the square root of 170 and when you solve it is approximately 13.04.

2y = -x + 6 2y = -x + 6 /2 /2 /2
 * 5.** Write the slope-intercept form of the equation of the line satisfying the given conditions: the line passes through (-2, 4) and has a slope of -1/2. (lesson 8.3) __**All of my work is below!**__
 * Step 1 Write in Point-Slope Form:** x + 2y = 6
 * Step 2:** Subtract x
 * Step 3:** Divide by 2
 * Final Answer:** y = -1/2 + 3

(All of my images were found on [] All credit goes to them for the pictures!)

Taylor Carroll - April 29, 2011

 * 1.** I feel that Kepler was a little of both. This is because with everything that he researched scientifically he supported that research with mathematics. He would research the planets and then mathematically compute when farmers should plant their crops for the best results. Though this scientific and mathematical give and take Kepler helped solve and support every problem someone could throw at him. The conic sections had a very large part in Kepler's planetary research. They had their first real application when Kepler submitted his first Law Of Planetary Motion. Through the use of conic sections Kepler determined that the planets move in a elliptical pattern.

Kepler developed methods of calculating sizes of areas with curved lines. He submitted the first proof of how logarithms work. He also developed a number system based on Roman numerals. His contributions to mathematics are still very important today because without his research we would not know how to calculate the sizes of areas with curved lines, we wouldn't have his knowledge of logarithms and we would not have his Roman-like number system. He also helped further scientific and mathematical research that has helped society advance in many ways.


 * 2.** Logarithms can be used in many different ways in the real world! The first way they can be used is in the Richter Scale. The formula for the Richter Scale is R = log[ I / I0 ] and I0 represents the "threshold quake". The threshold quake is the movement in the ground that can be barely felt. I represents the intensity of the quake and is represented in multiples according to the threshold quake.

The next way logarithms can be used in the real world is measuring radioactive decay. The formula for radioactive decay is shown below:

Where M= the mass' original value, t = time, and h = the half-life. The half-life is the amount of time it takes for the substance to one-half of it's original value.

The last way logarithms can be used in the real world is in measuring the decibels of sound. The formula for this is below:

Where I = the intensity. This makes it easy for scientists to calculate close to the exact decibels of sound an object makes.


 * 3.** A type of problem that gave me issues in the beginning of chapter 8 was finding the slopes of lines, but once I memorized the slope formula I no longer had issues with these types of problems. Here is an example of a "find the slope" problem.

Find the slope of the line passing through the points (6,4) and (-1,2)

To solve this you have to start with the slope formula: Where M is the slope. You must then take the y and x values and plug them into the slope formula. So, you would take y2 (2) and y1 (4) and subtract them according to the formula. 2-4 = -2 After, you would take the X2 (-1) and X1 (6) and subtract them according to the formula. -1-6 = -7. Lastly, you must take -2 and put it over -7 which gives you -2/-7, the negatives cancel out and you are left with 2/7. So, M = 2/7!


 * 4.** I used the system x + y = -1 and x - y = 7. So, I set it up like this:

x + y = -1 x - y = 7

After, I used the elimination method to solve this problem. This gave me 2x = 6, then I divided by 2 and I got x = 3. Lastly, I plugged x = 3 into both equations to get 3 + y = -1 and 3 - y = 7. The solution to these problems is -4. The final solution is (3,-4). This is a consistent and independent system.


 * 5.** I would first tell them what a system of inequalities is, so they could understand what they were solving. A system of inequalities is a set of linear inequalities that you are solving all at one time. Then, we would look at the problem and determine how to solve it. For example:

Solve the following system:

2x - 3y __<__ 12 x + 5y __<__ 20 x > 0

Then, we would solve for y in the first two equations. The best way I can explain this would be to "switch" the x to the other side. Once we do that we get the following system:

y __>__ (2/3)x - 4 y __<__ (-1/5)x + 4 x > 0

To solve the system we have to graph them and the overlapping portion of the graph is the solution. So, lets graph //y// __>__ ( 2/3 )//x// – 4 first! When we do that we get a graph that looks like this:



Then, we have to determine whether the problem has the symbol greater than or less than.

y __>__ ( 2/3 )x – 4

It is a greater than inequality, so that means we are going to shade above the line. We don't want to shade the whole graph though because we haven't the second inequality! Just remember, we are shading ABOVE the line for the first equation.

Now that we are finished graphing the first inequality we can move on to the next one!

y __<__ ( – 1/5 )x + 4

When we graph this equation it looks like this:

Now we once again have to determine whether this is a greater than or less than problem.

This is a less than problem, so that means we are going to shade BELOW the line.

Now that we have our inequalities graphed we can find the solution! The solution is everything between the two lines, but remember x has to be greater than zero. No negatives! The solution to the inequality is every area where the inequalities are satisfied and there you have it, how to solve systems of linear inequalities!



(I got the problem above from []. I typed my own explanation, but I used their graphs and problem. All credit for that goes to them!)

Website sources:

[] [] [|http://www.uiowa.edu/~examserv/mathmatters/tutorial_quiz/log_exp/realworldappslogarithm.html] [] [] []

May 13, 2011 - Taylor Carroll
**1.** The Greeks were very adept in Geometry and they studied it solely as their form of mathematics. They did not look for the practical applications of geometry because it was all abstract to them. They saw abstract beauty in Geometry and overlooked everything else. They had two tools for Geometry, compasses and straightedges. One of their most important rules was that the straightedge should have no measurements on it, much like Mrs. Sivits told us at the beginning of our construction packets. The Greeks were also not allowed to line up the points by eye.

The Three Famous Problems of Antiquity:


 * 1. To trisect an arbitrary angle.
 * 2. To construct the length of the edge of a cube having twice the volume of the given cube.
 * 3. To construct a square having the same area as that of a given circle.

It was very interesting to find out that these famous problems would not be accomplished with the Euclidean tools. They were eventually solved using other methods, but these methods violated the Greek Geometry rules.

This article was extremely interesting because you receive a greater understanding of the Greeks' form of mathematics. I learned that they did not see the practical applications as something great, it was just an added bonus to the beauty of the shapes. The beauty of the shapes was very important to the Greeks because it signified an abstract style that could only be achieved through geometry.

**2.** There are many different ways shapes can be used in the real world. Here are five fun examples!

Octagon: This is probably the most obvious of all the shapes and probably the most ignored shape in the whole world. Yes, I am talking about the stop sign. The octagon is the sole shape for the stop sign, no other road sign can be an octagon. So, next time you go out driving look for the octagon and remember to STOP!

Rectangle: This shape is the basic shape for most of the books we read everyday! The fun rectangular shape makes it easy for you to hold and read!

Circle: This shape can be associated with the tire! The circular shape is what makes our wheels have the ability to roll, what would our vehicles be like if there were no circular wheels?

Triangle: This is another shape that is frequently seen on the road. There are yield signs, warning signs, and slow vehicle signs. The most common triangular sign seen here in Knox County is the slow vehicle sign! These signs are most commonly put on the back of Amish buggies and farming equipment! This warns other drivers that the "vehicle" in front of them is slow moving and to be careful when passing, turning or driving around these "vehicles".

Pentagon: This shape is used in our nation's department of defense headquarters. Yes, I am talking about the Pentagon; The Pentagon is obviously named for it's pentagon shape. This is the most famous use of the pentagon shape.

**3.** Euclid contributions to mathematics were very important to geometry. He has been called the "Father Of Geometry" because he wrote The Elements, which defined geometry as we know it. The "series" of The Elements had thirteen books in total. These books had the teachings of mathematicians like Pythagoras, Hippocratus and many other famous mathematicians. These books also included Euclid's own views and teachings on geometry.There was one reason why Euclid's Elements was so popular, that reason was simplicity. The way Euclid described each geometrical subject was very easy to understand and it made mathematics more available to different types of people.

Euclid's Geometry has been studied for twenty three centuries and it has shaped geometry into what we use today. Euclid's geometry has been challenged though and many modern mathematicians have found "holes" in Euclid's The Elements. Euclid has also written many other books on mathematics such as, Data, Division Of Figures, Phenomena, and Optics. His teachings are still being studied today and people continue to learn from them.

If I could talk with Euclid today I would probably tell him that his greatest work The Elements has been challenged and some of his teachings have been proven wrong. I would then ask him if he would be able to fix his mistakes and once again prove his geometry masterpiece.



**4.** A rhombus is the "offspring" of a parallelogram and it must have all of it's sides equal in length. A parallelogram is the "parent" of a rhombus and is defined as a quadrilateral with opposite sides parallel. So, while a rhombus must be a parallelogram because it is from that family, a parallelogram has many other "offspring" and can be something other then a rhombus.

__**Books and websites credited here:**__


 * Mathematical Ideas - Miller, Heeren, and Hornsby**

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[|http://www.gap-system.org/~history/Biographies/Euclid.html]

[|http://www.gap-system.org/~history/Mathematicians/Euclid.html]

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