...2a. General Equation of the Ellipse From the general equation of all conic sections, A and C are not equal but of the same sign. Thus,the general equation of the ellipse is Ax2 + Cy2 + Dx + Ey + F = 0 or Standard Equations of Ellipse From the figure above, and From the definition above, Square both sides Square again both sides From triangle OV3F2 (see figure above) Thus, Divide both sides by a2b2 The above equation is the standard equation of the ellipse with center at the origin and major axis on the x-axis as shown in the figure above. Below are the four standard equations of the ellipse. The first equation is the one we derived above. Ellipse with center at the origin Ellipse with center at the origin and major axis on the x-axis. Ellipse with center at the origin and major axis on the y-axis. Ellipse with center at (h, k) Ellipse with center at (h, k) and major axis parallel to the x-axis. Ellipse with center at (h, k) and major axis parallel to the y-axis. The Hyperbola Submitted by Romel Verterra on February 21, 2011 - 1:33pm Definition Hyperbola can be defined as the locus of point that moves such that the difference of its distances from two fixed points called the foci is constant. The constant difference is the length of the transverse axis, 2a. General Equation From the general equation of any conic (A and C have opposite sign, and can be A > C, A = C, or A < C.) ...
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...the Ellipse From the general equation of all conic sections, [pic] and [pic] are not equal but of the same sign. Thus, the general equation of the ellipse is [pic] or [pic] Standard Equations of Ellipse Elements of the ellipse are shown in the figure above. 1. Center (h, k). At the origin, (h, k) is (0, 0). 2. Semi-major axis = a and semi-minor axis = b. 3. Location of foci c, with respect to the center of ellipse. [pic]. 4. Length latus rectum, LR 5. Consider the right triangle F1QF2: Based on the definition of ellipse: [pic] [pic] [pic] By Pythagorean Theorem [pic] [pic] [pic] [pic] [pic] [pic] [pic] [pic] [pic] [pic] You can also find the same formula for the length of latus rectum of ellipse by using the definition of eccentricity. 6. Eccentricity, e DEFINITION: Eccentricity of Conic Eccentricity is a measure of how much a conic deviate from being circular, making the eccentricity of the circle obviously equal to zero. It is the ratio of focal distance to directrix distance of the conic section. [pic] From the...
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...Earth, People, and Environment Exam 1 Exam 1 covers Chapters 1-6 in the textbook, all lectures, and homework assignment 1. Chapter 1: Thinking Geographically What does the term Geography literally mean based on its Greek roots? Why are natural hazards a good example of the issues geographers study? What are the tools of GIS, GPS, Google Earth, Google Maps, mashups, and remote sensing used for? What is spatial analysis or geographic thinking? What are these terms geographers use to describe the earth? place, region, scale, space What is cartography? Is it an art or science, both? What are some ways of representing map scale? Why are Map projections needed? Why is there always some map distortion or errors in maps? What kind of map projection is the Mercator projection: cylindrical, conic, azimuthal? What areas of the globe are best portrayed by a conic projection? What are lines of latitude and lines of longitude? Where does the Prime Meridian run through? What is Greenwich Mean Time? Where is the International Date Line? How do we refer to places using latitude and longitude and the geographic grid (e.g., GPS locations?) If a football game begins at 1 pm EST, what time does it begin in Los Angeles? How are site and situation related? What is a vernacular region? Functional region? What are some pros and cons of globalization on culture in different countries? What does globalization refer to? What is culture? Is it dynamic or static? Chapter 2: Weather, Climate, and Climate Change...
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...9.1 Circles: 1. Write the equation in standard form for the circle with center at and radius 2. Write the equation in standard form for the circle with center at and radius 3. Graph the circle given by 4. Graph the circle given by 5. Write the equation of the circle in standard form given: 9.1 Parabolas: 6. Find the focus of the parabola 7. Find the focus of the parabola 8. Write the equation of the parabola in standard form and find the focus and directrix. 9. Write the equation of the parabola in standard form and find the focus and directrix. 10. Write the equation for the parabola with vertex and focus 11. Write the equation for the parabola with vertex and directrix 9.2 Ellipses: 12. Identify the center, vertices, & foci of the ellipse given by and graph. 13. Identify the center, vertices, & foci of the ellipse given by and graph. 14. Write the equation in standard form: 9.5 Parametric Equations: 15. Write and in rectangular form. 16. Write each pair of parametric equations in rectangular form: 17. Write and in rectangular form. 18. Write and in rectangular form. 9.6 Polar Equations: 19. Graph the following polar coordinate: 20. Graph the following polar coordinate: 21. Graph the following polar coordinate: 22. Graph the following polar coordinate: 23. Find the polar coordinate of the point 24. Find the polar coordinate of the point 25...
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...MA1200 Basic Calculus and Linear Algebra I Lecture Note 1 Coordinate Geometry and Conic Sections υ MA1200 Basic Calculus and Linear Algebra I Lecture Note 1: Coordinate Geometry and Conic Sections Topic Covered • Two representations of coordinate systems: Cartesian coordinates [ሺݕ ,ݔሻcoordinates] and Polar coordinates [ሺߠ ,ݎሻ-coordinates]. • Conic Sections: Circle, Ellipse, Parabola and Hyperbola. • Classify the conic section in 2-D plane General equation of conic section Identify the conic section in 2-D plane - Useful technique: Rotation of Axes - General results φ MA1200 Basic Calculus and Linear Algebra I Lecture Note 1: Coordinate Geometry and Conic Sections Representations of coordinate systems in 2-D There are two different types of coordinate systems used in locating the position of a point in 2-D. First representation: Cartesian coordinates We describe the position of a given point by considering the (directed) distance between the point and -ݔaxis and the distance between the point and -ݕaxis. ݕ 0 ܽ ܲ ൌ ሺܽ, ܾሻ ܾ ݔ Here, ܽ is called “-ݔcoordinate” of ܲ and ܾ is called “-ݕcoordinate” of ܲ. χ MA1200 Basic Calculus and Linear Algebra I Lecture Note 1: Coordinate Geometry and Conic Sections ܲଶ ൌ ሺݔଶ , ݕଶ ሻ ܲଵ ൌ ሺݔଵ , ݕଵ ሻ Given two points ܲଵ ൌ ሺݔଵ , ݕଵ ሻ and ܲଶ ൌ ሺݔଶ , ݕଶ ሻ, we learned that • the distance between ܲଵ and ܲଶ : ܲଵ ܲଶ ൌ ඥሺݔଶ െ ݔଵ ሻଶ ሺݕଶ െ ݕଵ...
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...In computing, mmap is a POSIX-compliant Unix system call that maps files or devices into memory. It is a method of memory-mapped file I/O. It naturally implements demand paging, because initially file contents are not entirely read from disk and do not use physical RAM at all. The actual reads from disk are performed in "lazy" manner, after a specific location is accessed. In Linux, Mac OS X and the BSDs, mmap can create several types of mappings. Anonymous mappings are mappings of that area of the process's virtual memory backed by the swap space instead of by a file in the file system name space. In this respect an anonymous mapping is similar to malloc, and is used in some malloc implementations for certain allocations. However, anonymous mappings are not part of the POSIX standard, though implemented by almost all systems. File-backed mappings are mappings of virtual memory to files. Access to those areas of memory causes the file to be read. If the mapping is shared, writes to that area in one process will affect other processes with that area mapped and the underlying file; otherwise, if the mapping is private, the changes will not be seen by other processes nor written to the file. A process reading from or writing to the underlying file will not always see the same data as a process that has mapped the file, since the segment of the file is copied into RAM and periodically flushed to disk. Synchronization can be forced with the msync system call. mmaping...
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...ARCHITECTURAL DESIGN 6 Design Plate number 3 “ PHILIPPINE/NATIONAL TRANSPORTATION MUSEUM” THE PROBLEM: THE Philippine Government thru the Ministry of transportation in cooperation with several civic organizations has proposed to construct Land Transportation Museum (Phase-1) to preserve and save all different types of land transportation in existence and value to show and document our heritage thru the evolution of land transportation in the Philippines. THE PURPOSE: There are a lot of cars and other vehicles of value and a few have already been restored by some club members of the Vintage Car Club of the Philippines Such as the V-16 Cadillac Limousine of Gen. Douglas MacArthur, the Packard Limousine of president Jose P. Laurel Sr. and lately the Chrysler Zypher, bullet proof car of president Manuel L. Quezon. These cars, all part of our transportation history, are kept in different place in our country. These priceless cars need a permanent home, where it can be shown to public at the same time, be a tourist attraction which would contribute immensely to the Tourist industry. THE SITE: The site shown in the last two pages is located at the Manila-Cavite coastal road, very close to the site of the sport complex and almost in front of the Asian Development Bank Building. The lot has an area of approximately 30,000 SM. The proposed area has a relatively flat terrain which slopes towards the 30,00 M. wide Estero which is adjacent to the Manila Bay. THE REQUIRED DRAWINGS: ...
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...centralized database pertaining only to the city and county of San Diego. What this does, is contractors, landscapers, and essentially anyone else who is inquiring about data regarding environmental or geographic information, can pay to access the system. For instance, by cooperating with other local agencies, SanGIS can uphold the integrity of geographic information with accuracy and precision for any party that is interested in statistical information, data extractions, data sharing, etc. For an example query of the application, someone could inquire on where the closest fire stations, schools, public libraries, police stations, etc. are, using an interactive map. There are multiple filters to the interactive map; a base map that shows roads, county lines, parcels, and city lines. A second filter to the interactive map is called a dynamic filter. The dynamic filter can be turned on and off by the user and has three categories; districts, ecology, and places. It also includes data such as vegetation and public service stations. The primary use would be if a company wanted to build a new building in a vacant lot, they would use SanGIS to figure out where the lot is in correlation to exact city, districts for emergency services, county lines for legislative purposes, as well as coordinates...
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...1. What were the driving forces that lead Wells Fargo OFS to redesign their strategy and processes? Wells Fargo launched its online financial services much ahead of their competitors and with the astonishing response of the growing customer base, they enhanced the online services with all major banking services like reviewing their accounts and balances, bill payments etc. The OFS primarily focused on the maintaining the high attrition rate of online customers and provide all their customer with a low cost service medium. The OFS group recovered their cost by charging other department within the bank depending on the activity of the customer after logging into the portal. However due to this main the purpose of providing a low cost media to the customer was being lost in the need of generating OFS group revenue. Because of this the OFS had reached a point where the profits and costs were equal. Wells Fargo’s traditional approach to measure performance and strategies using the financial metrics led to poor decision making while the online banking needed a well-planned long term goal as it was one of the driving factor of the business in the coming years. Also, because of the rapidly changing technological trends on a daily basis in the internet media, the OFS had to be upstream in their approach to keep with the latest technologies. This involved new competitive threats and meeting the customer demands became a challenge due to the varied customer base in the online banking...
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...influence one another over time; and why places and the people who live in them develop and change in particular ways. Raising these questions is at the heart of the “geographic perspective.” Exploration has long been an important part of geography. But exploration no longer simply means going to places that have not been visited before. It means documenting and trying to explain the variations that exist across the surface of Earth, as well as figuring out what those variations mean for the future. The age-old practice of mapping still plays an important role in this type of exploration, but exploration can also be done by using images from satellites or gathering information from interviews. Discoveries can come by using computers to map and analyze the relationship among things in geographic space, or from piecing together the multiple forces, near and far, that shape the way individual...
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...that function well, as well as areas that we investigated and had concerns about. There is a level of reasonableness that goes into safety but accidents are not planned and in an emergency situation they are the risk of one is higher. An example of a emergency could be a fire, earthquake, power outage, flood, or threat. The purpose of addressing these concerns is to help prevent unnecessary accidents for staff and student. The building our team looked at was the Library and upper study hall by the library. Building Features The buildings were built to a certain level of safety and it is important to address the positives that already exist. * Sprinklers cover all area of floor space so fires would struggle to spread * Building maps located throughout the school that highlight exits and fire extinguishers * Easily spotted and accessible fire extinguishers * Stairs have added tread to avoid slipping for normal walking or emergency evacuation * Light fixtures have protective cover, so glass will never fall from ceiling Safety Oversights The following include safety concerns that arose concern when considering what could happen in an emergency situation when accidents happen. * When the building was first designed all the chair in the upper study by the library were bolted down away from the railing but over time more chairs have been added and those chairs are the same level as the railing. It is a full story fall down from that level and currently...
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...retreat (15 marks) Fieldwork is the collection of primary data carried out at the location in the field. Research is secondary data and archival research normally carried out before or after investigation in the field. Coastal retreat is the removal of sediment from the shoreline/erosion resulting in the coastline moving back. During the investigation it would be important to consider the different fieldwork technique that I have used e.g. GOAD mapping this technique is a record of different business and building in the location however this data is being collected by walking around the location looking at GOAD map to see changes overtime, this appropriate and out as evidence because if there was building on the coast that is not present it show in comparison that the coast has been retreat. This information may not be accurate because of the limitation has in the GOAD map may not have enough details about building, house and business details. The second fieldwork technique that I would undertake is field sketch/photos this is being collected by drawing a sketch of landmarks and geologically feature of the land like cliffs, headland and wave cut platform, the more detail being draw the more accurate the field sketch also the photos are being taken by a camera to record the feature of the location in more details. This may act as evidence for coastal retreat because if you compare other field sketch that was done by other researcher and there have being changes in the land mass...
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...Geography - Maps Continents and Oceans of the World There are 7 continents in the world 1. Africa 2. Antarctica 3. Asia 4. Australia 5. Europe 6. North America 7. South America There are 5 Oceans 1. Pacific 2. Atlantic 3. Southern 4. Indian 5. Arctic The Pacific Ocean is the largest of the world's five oceans, followed by the Atlantic Ocean, Indian Ocean, Southern Ocean, and Arctic Ocean. The ocean covers 71 percent of the Earth's surface and contains 97 percent of the planet's water. Map skills To use a map effectively you need to know how to read the information. This is done by learning basic map skills. Measuring Distance Being able to measure the distance between two points on a map is very important. It allows you to work out what distance is in real life. Every map is printed with a scale bar that converts the distance you measure on a map (usually in centimetres or inches) into a real life distance (usually in kilometres or miles). A quick way to measure distance is to count each square you cross on the map. Understanding Scale It wouldn't be possible for maps to show things the size they are in real life, so maps make things smaller using scale. Drawing something to scale means showing it at a different size to what it is in real life. When maps are drawn to scale things are made many times smaller than they really are. Because maps are important to a lot of people, this...
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...Question 2 Map scale is the relationship between the distance on a map and the distance in reality. Map scales can be represented as a scale ratio, a word statement or a graphic line scale. Example of a scale ratio would be 1:2500 and the word statement would be 1 cm on the map equals 25m on the ground, while a line scale is a line on the map representing distance in reality which can be measured to determine the scale. A large representational scale would be a large fraction like 1:2500, while a small scale would be a smaller fraction like 1:1000000. The larger the scale the closer to reality is...
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...Running head: APPLICATIONS OF A CONIC SECTION Applications of a Conic Section and its use in Mathematics Shabue C. Johnson ITT Technical Institute APPLICATIONS OF A CONIC SECTION A conic section is the curve made when you take a particular angle from a double napped cone. A double napped cone is one cone placed on another and the tips of the cones match up exactly with the tips lined up perfectly. When taking a “ Slice” through the cone you are actually taking a section of the cone and from that section you are dealing with a shape. If you take an exact slice through the cone horizontally, you are left with a circle. If you take a slice roughly at a forty five degree, you will be dealing with an ellipse. If you take a slice that is parallel from one edge of the cone to the other cone, you are dealing with a parabola. If you take a slice from directly off centered but straight down from top to bottom, you give yourself a hyperbola. These are a few terms with definitions you will see while working with conic sections. In a circle, ellipse, and a hyperbola you have a Center. Which is usually at the point of (h,k.) The focus or “Foci” is the point which distances are measured in forming the conic. The directrix is the distance that is measured in forming the conic. The major access is the line that is perpendicular to the directrix that passes through the foci. Half of the major axis between the center and the vertex is called the semi major access. There is a general...
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