Conic Sections C I R C L E S Conics can be found all around us; whether it be in architecture, or in everyday items. Throughout history, conics have helped build some of the most amazing buildings around the world. According to Encarta World English Dictionary, conic sections are curves produced by the intersection of a plane with a circular cone, e.g. a circle, ellipse, hyperbola, or parabola. Even though we may not notice conics, they are in almost everything we see. Conic sections are among the oldest of mathematics. It is believed that a Greek mathematician name Menaechmus discovered conic sections. He was also the tutor of Alexander the Great. Another mathematician named Appollonius took Menaechmus’ discoveries and refined it to create a more extended version of the conics that Menaechmus had discovered. Appollonius not only based the theory of conics off of one cone, he also named the conics: hyperbola, ellipse, and parabola. During the renaissance, conic sections were pushed to a higher level by many mathematicians of that time period. They contributed to the conics and also had some mathematical advancements of their own such as, Kepler’s Law of Planetary motion. Circles are almost everywhere because they are the most useful and most obvious to find. The formula of a circle isx^2+y^2=a^2 . The formula may also be written asx^2/a^2 +y^2/b^2 =1. Sometimes, circles and ellipses might be mistake for one another because their equations are very similar. To differentiate a circle from an ellipse, you must check ifa^2is equal tob^2. If they are equal, then it is a circle, and id they are not equal, then it is an ellipse. We use circles to make wheels for land transportation, e.g. cars, bikes, trains, skateboard, and other forms of transportation. Without circles, land transportations would not be possible. Circles are also used in everything from everyday household items to anything you can think of. Circles are in design, technology, food, and clothing. Circles may just be one of the most useful conics E L L I P S E S Ellipses are also found almost everywhere. The formula of an ellipse isx^2/a^2 +y^2/b^2 =1. It is very similar to a circle; the only difference is the a^2and the b^2are different. The x^2and the y^2are what determine whether the ellipse will be horizontal or vertical. The earth and the moon have an elliptical orbit as well as the satellites that orbit the earth. Before the 17th century, many Greek astronomers believed that the earth was the center of the universe and the all of the planets revolved around it in a circular motion. This, of course, was proven to be wrong by an astronomer name Johannes Kepler. He believed that the planets travelled around the sun in a elliptical orbit, rather than a circular one. Since this was discovered during this time period, it was very difficult for people to believe that this information was true and so many scientist and astronomers were accused of heresy. Another conic is a parabola. The equation for a parabola is y=x^2/4a. A parabola can face up or down, and right or left. This conic is not as common as an ellipse or a circle. Parabolas can also be found in architecture and in places you would never expect. They can also be found in physics and in everyday things. Galileo discovered the parabolas when trying to design a more efficient way of shooting cannons. He wanted to teach the people aiming the cannon how to make it travel a certain path by aiming it a certain way. Parabolas can also be found when a golf ball or a baseball is hit. These two balls follow a path similar to a parabola. Parabolas are used in satellites and antennas. One famous place where you can find a parabola is in the Bellagio in the Las Vegas, Nevada. The fountains in front of this luxurious hotel and casino are shaped in a parabola form. When the show starts, the fountains shoot up into the air in a parabolic line. In architecture, parabolas are use in designing arcs and such. Hyperbolas are not as common as the rest of the conics. The equation for a hyperbola isx^2/a^2 -y^2/b^2 . As you can see, a hyperbola equation has a minus sign instead of a plus sign. This is one unique way to distinguish between a hyperbola and an ellipse or parabola. The parabola is a little harder to find in everyday life. You can surely see it in architecture. Many buildings that have curves are made out of parabolas.

P R A B O L A S A H Y P E R B O L A S This project has helped us learn how conic sections are found in almost everything and everywhere. Before this project, we did not realize how important conics are in everyday life. This project has also taught us the history of conic sections and who they were discovered by. During this project, we couldn’t really meet up to work on this together. Instead, we split the work and made it work. Our schedules are the same and we really didn’t have time to work on this much. We did our best and we hope that our best will be enough. The places that we photographed are places that Denisse has been to and had photographed before. She used to take photography classes and so she learned about cool places to take pictures at. I think that for next year, you should give a bit more of time. Or, maybe you should assign this during spring break so everyone has more time to work on it without having to work on other homework. This project has been fun and difficult. We really had fun during this project. CONCLUSION

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