Did you know that the Olympic torch is lit months before the games begin? The ceremony for igniting the flame is the same as it was in ancient times. The ritual is held in the Greek Temple of Hera in Olympia. The ceremony, which is based on Greek mythology, honors Prometheus, who stole fire from Zeus and gave it to humanity. The torch is placed in the center of a parabolic mirror, which focuses sunlight and ignites the flame.
Despite its association with Greek mythology and Olympic history, conic sections in mathematics, such as the parabola, were not invented. Instead, they were found, examined, and applied. Conic sections are created by crossing a plane with a cone. A cone is composed of two equally formed portions known as nappes. A nappe is what most people mean when they say cone ⏤ it’s shaped like a party hat!
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A parabola is a continuous curve that resembles an open bowl with sides that continue to rise indefinitely. It is made from a set of points that are all the same distance apart from a fixed point called the focus, and a line called the directrix. A parabola has multiple uses in mathematics and science and is employed in many practical applications today.
A parabola can also be defined as a specific conic section. A parabola can be seen if you slice parallel to one side of a cone. When the parabola curve is symmetrical around the y-axis, it is characterized by the equation y = ax2 + bx + c, although there are more generic equations for other circumstances.
Parabolic mirrors (or reflectors) can trap and focus energy on a single point. The parabolic gadgets used daily to demonstrate the benefits of this property include:
- Satellite dishes
- Suspension bridges
- Automobile headlights
Parabolic reflectors are often utilized in alternative energy equipment such as solar cookers and water heaters since they are inexpensive to manufacture and require minimal maintenance.
Parabola in Real Life
Parabolas can be found everywhere, in nature and in artificial objects. Consider a fountain that shoots water into the air and returns in a parabolic route. A ball thrown in the air also follows a parabolic route, as Galileo demonstrated. Anybody who has ridden a roller coaster is familiar with the rise and fall caused by the track’s parabolas. Some other examples of parabolas in real life are:
Architectural and Engineering Parabolas
Parabolas are used in architectural and engineering applications as well. The Parabola, a London structure built in 1962, has a copper roof with parabolic and hyperbolic lines. The iconic Golden Gate Bridge in San Francisco, California, features parabolas on either side of its side spans or towers.
Whenever light must be focused, parabolas are widely employed. Lighthouses have undergone several modifications and enhancements to achieve the quality of light they can generate today. Flat surfaces scatter too much light to be beneficial to seafarers. Spherical reflectors improve brightness but do not provide a strong beam. Using a parabola-shaped reflector helps to focus light into a beam that can be seen from a considerable distance.
The first documented example of parabolas being used in lighthouses was the use of parabolic reflectors to build a lighthouse in Sweden in 1738. Over time, several other forms of parabolic reflectors would be introduced to decrease lost light and increase the surface of the parabola. Glass parabolic reflectors were eventually used when electric lights were introduced, and the combination proved to be an effective method of transmitting a lighthouse beam.
Another example of parabolas being used in real life is their use in headlights. From the 1940s to the 1980s, sealed-beam glass automotive headlights employed parabolic reflectors and glass lenses to focus light beams from bulbs, improving driving vision. Later, more efficient plastic headlights were molded to eliminate the need for a lens. These acrylic reflectors are now widely used in headlights.
The solar power sector is increasingly benefiting from the use of parabolic reflectors to concentrate light. Flat photovoltaic systems collect light and release electrons from the sun, but do not concentrate them. A curved photovoltaic mirror, however, can concentrate solar electricity considerably more efficiently.
A great example of using parabolas in real life is the Gila Bend parabolic trough solar installation, Solana, which is made up of massive, curved parabolic reflectors. In this solar-powered system, the parabolic mirror focuses sunlight to create heat. This warms synthetic oil tubes in the trough of each mirror, generating steam for electricity, or storing the energy in large pools of molten salt for later use. This use of a parabola in real life allows more energy to be stored and produced, making the process more efficient.
Perhaps the most remarkable illustration of a parabola in real life is the shimmering, stretched arc of a rocket launch. When a rocket or other ballistic item is launched, it follows a parabolic path, also known as a trajectory. This parabolic trajectory has been employed in spaceflight for decades.
Specialized airplanes that fly at a steep angle provide a higher-gravity sensation before dropping into freefall, resulting in a zero-gravity experience. Chuck Yeager, an experimental pilot, tested the use of parabolas in flight to help understand a person’s tolerance for spaceflight and their ability to fly in different gravities. Parabolic flights save money by allowing these tests to be done on Earth rather than in space, thus avoiding the need to conduct every experiment in space.
A hyperbola is a smooth curve within a plane and is described by its geometric qualities or by equations for which it is the solution set. A hyperbola is made up of two portions known as linked components or branches, which are mirror images of one another and resemble two infinite bows. If a plane overlaps both portions of a double cone but does not pass through the apex of the cones, the conic is a hyperbola. A hyperbola is one of three types of conic sections formed by the intersection of a plane with a double cone. The parabola and ellipse are the other types of conic sections.
Examples of Hyperbola in Real Life
When an aircraft travels faster than Mach 1 (the speed of sound), it forms a cone-like wave that illustrates a real-life example of hyperbolas. Where the cone hits the ground, it is a hyperbola.
At the same time that the hyperbola is forming, a sonic boom strikes every point on the arc. Almost no sound is heard outside of the bend of the hyperbola. The sonic boom curve is named after the hyperbola because of their similarities.
Nuclear Reactor Cooling Towers and Coal-fired Power Plants
All nuclear cooling towers and a majority of coal-fired power facilities use the hyperboloid design, a perfect display of hyperbola use in real life. They have a solid structural foundation and can be erected with straight steel beams.
Difference Between Parabola And Hyperbola
A parabola is a group of points in a plane that are equidistant from a straight line, directrix, and focus. A hyperbola is defined as the difference in distances between a collection of points in a plane and two fixed points that are positive constants.
An ellipse is a plane curve that goes around two focus points. It has a constant sum of the two distances to the focal points at all places on the curve. And so, it generalizes a circle, which is a specific form of an ellipse with the same two focus points. There are multiple applications of conics in both pure and practical mathematics. Planetary and satellite orbits are ellipses. Ellipses are also employed in the manufacture of machine gears.
Examples of Conic in Real Life
The ellipse is the most common application of a conic curve encountered in everyday life. Here are some examples of where you can see an ellipse:
- When you tilt a glass of water, the surface of the water takes an oval appearance, forming an ellipse.
- Salami is normally cut obliquely to produce elliptical slices.
- The orbit of the Earth’s artificial satellites, the moon, and the routes of comets that orbit the sun are elliptical.
- The Statuary Hall inside the U.S. Capitol Building is an elliptical construction.
- An elliptical billiard table exhibits the ellipse’s capacity to bounce an item from one focus to another, allowing a ball to rebound to another focus when positioned at a specific focus and shoved with a cue stick.
These examples of parabolas, hyperbolas, ellipses, and conics in our everyday lives can help you understand these mathematical concepts. If you wish to continue your math journey, sign up for a FREE trial class at BYJU’S FutureSchool, where you will be taught in a 1:1 environment by a live instructor. You can also read more math resources on BYJU’s FutureSchool blog.