9.1.2 – applications of ellipses, other information
TRANSCRIPT
9.1.2 – Applications of Ellipses, Other Information
• We have so far discussed the basics of an ellipse– Specific type of conic section– “Oval”– Major vs Minor Axis– Locations of foci– Graphing; centered at origin or somewhere else
Skinny Factor
• As mentioned yesterday, all circles are, in fact, a form of an ellipse– Major axis length = minor axis length
• A general rule of thumb is that a narrow ellipse will have two foci far away from the center
• Those that are nearly circle will have foci relatively close to the center
• To help us quantify a “skinny factor,” we define the eccentricity of an ellipse as;
•
• If e is close to 1, then you have a narrow (skinny) ellipse
• If e = 0, then it is a circle
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22
• Example. Find the eccentricity and the lengths of the minor and major axes for the ellipse
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• Example. Find the eccentricity and the lengths of the minor and major axes for the ellipse
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• Example. Find the eccentricity and the lengths of the minor and major axes for the ellipse
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• Example. Find the eccentricity and the lengths of the minor and major axes for the ellipse 252 22 yx
Planets/Solar System
• Johannes Kepler first demonstrated that planets follow elliptical orbits
• With his work, we have Kepler’s Three Laws of Planetary Motion
• 1) Planets orbit the sun in elliptical paths, with the sun as a focus of each orbit
• 2) A line segment between the sun and given planet sweeps over equal areas of space in equal time intervals
• 3) The square of time needed for a planet’s revolution about the sun is proportional to the cube of half of the major axis
• Example. The furthest Earth gets from the sun is 94.56 million miles, and the eccentricity is about 0.017. Estimate the closest approach of the Earth to the sun.
• Example. The orbit of Halley’s comet is an ellipse with an eccentricity of 0.967. It’s closest approach to the sun is about 54,591,000 miles. What is the furthest Halley’s Comet gets from the sun?
• Assignment• Pg. 709• 53-61• 63, 64, 65
