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Real World Conic Sections!
Transcript of Real World Conic Sections!
Anthology of Art and Architecture of European Significance
You can find
When Relaxing and getting a facial, they use circular cucumbers to cover your eyes. It's part of the facial and helps with dark circles and puffiness under the eyes.
Real World Conic Sections!
by Kristy Vickers
One of Nature's Awesome Creations: The Hurricane
The roman Antiquity of the Classical period has left and indelible mark on the City of Pula, where today we are still able to admire the magnificent Amphitheater.
All the conic sections can be created by passing a plane through a double-napped cone and looking at the cross section.
We can find Parabolas even in things that we enjoy doing!
Oh Yea ! It is time to have some fun riding on these.
"The parabola is the form taken by the path of any object thrown in the air, and is the mathematical curve used by engineers in designing some suspension bridges. The properties of a parabola make it the ideal shape for the reflector of an automobile headlight."
To the St. Louis Arch
A plane curve formed by the intersection of a right circular cone and a plane parallel to an element of the cone or by the locus of points equidistant from a fixed line and a fixed point not on the line.
Some where over the Rainbow!
Standard form of the equation for a Parabola.
Catch the wave!
a closed plane curve consisting of all points at a given distance from a point within it called the center. Equation: x ^2 + y^ 2 = r^ 2
One prime example of a circle that you can find in real life is a Ferris Wheel. All the points along the outer rim of the wheel are equidistant from the center. The lights on this one can help you see that a little easier.
They are the connections to places that we might not otherwise be able to get to unless by boat or ferry.
Without these everything would be dark.
are everywhere we turn.
In our homes, in nature, and architecture.
the set of points in a plane whose distances to two fixed points in the plane have a constant difference; a curve consisting of two distinct and similar branches, formed by the intersection of a plane with a right circular cone when the plane makes a greater angle with the base than does the generator of the cone. Equation: x 2 /a 2 − y 2 /b 2 = ±1.
The hyperbola is the least well understood of all the conic sections. Usually the illustration of a hyperbola and its relation to a cone shows two cones joined at the vertex with a plane cutting through both, parallell to the axis of the cone. Algebraically a hyperbola is like an ellipse (remember an ellipse is the set of all points the sum of whose distance from two points remain constant) only instead of the sum of such points, in the hyperbola, the difference of the distances remain constant.
The Standard Form of a hyperbola is very similar to that of the ellipse, except the two values are subtracted rather than added. The Standard form is x^2 / a^2 - y^2 / b^2 = 1 . As the value on the right side becomes larger than the value on the left side the difference between the two becomes negative, this is why the two sides face away from each other rather than towards each other as they do in the ellipse. Unlike an ellipse, the hyperbola is not a close figure so rather than a major axis and a minor axis, the hyperbola has a transverse axis outside the figure which is 2a units and a conjugate axis outside the figure which is 2b units long. The verticies are at (a, 0) and (-a, 0) and the foci are at (c, 0) and (-c, 0) just as with the ellipse with c being the distance from the focus to the center. The ends of a hyperbola do not keep expanding outwards as they do with a parabola, but limited, coming closer and closer to an imaginary boundary called an asymptote. The equations of these asymptotes are y = (b/a)x and y = -(b/a)x .
Cooling Towers of Nuclear Reactors
The hyperboloid is the design standard for all nuclear cooling towers. It is structurally sound and can be built with straight steel beams.
When designing these cooling towers, engineers are faced with two problems:
(1) the structutre must be able to withstand high winds and
(2) they should be built with as little material as possible.
The hyperbolic form solves both of these problems. For a given diameter
and height of a tower and a given strength, this shape requires less material
than any other form. A 500 foot tower can be made of a reinforced concrete
shell only six or eight inches wide. See the pictures below (this nuclear power
plant is located in Indiana).
A household lamp casts hyperbolic shadows on a wall.
Hyperbolic shadow cast by the sun and the tree. This creates an hour glass shape. The beauty of nature is you never know what you might find.
a plane curve such that the sums of the distances of each point in its periphery from two fixed points, the foci, are equal. It is a conic section formed by the intersection of a right circular cone by a plane that cuts the axis and the surface of the cone. Typical equation: ( x 2 / a 2 ) + ( y 2 / b 2 ) = 1. If a = b the ellipse is a circle.
This one is for all you sports fans!
What a weird looking plane.
In Belarus they use planes with ellipse wings. They say it has a few benefits comparing to the simple one or double winged planes, like the wing can be less in size, it’s more firm because the ellipse form is self sustaining, also there are now air vortexes by the sides of the wings which gives up to 30% increase in power compared to the traditional planes.
Why study ellipses?
Orbiting satellites (including the earth and the moon) trace out elliptical paths.
The Ellipse—with the White House and downtown Washington, DC, stretching out behind it—is one of the federal reservations. (Photo by Shutterstock.)
The Washington DC Ellipse, also known as the President’s Park South, is an ellipse located in front of the white house. The permanent Christmas tree that was planted in 1978 is lit by the president every year http://www.visitingdc.com/neighbor/washington-dc-ellipse.htm. The exact measurements of this ellipse are not given, however an approximate guess would be 300 feet in length and 200 feet in height. This gives a, the distance from the center to the end horizontally, a value of 150 feet and b, the distance from the center to the end vertically, a value of 100 feet. The standard equation of this ellipse, if the center is considered the origin, is:
x^2 / 22,500 + y^2 / 10,000 = 1
Real-Life Examples - Fun With Ellipses
This looks like it would be fun!
Hyperbolas, Ellipses, Circles, Parabolas
google images: equations for the conic sections