How A Parabola Is Used In Real Life gq5j

Ishan Gosain 11-7-12 The Future of Satellite Dishes Several companies today utilize satellite dishes to give their customers’ the best possible television experience. These companies, like Dish Network and Direct TV, are growing rapidly across the U.S. A study shows that 1 in 4 homes have satellite. Why are these companies flourishing? They’re success relates to their utilization of a parabolic shape for their satellites. If we want our company to succeed in the television market, then we need to take the satellite approach. The qualities of a parabola allow satellites to focus electromagnetic energy sent off from the company into space, or the energy sent to a person’s home. Also, its parabolic shape allows the waves from the satellites to travel further distances in contrast to spherical reflectors that suffer from spherical aberration (Wikipedia: The Free Encyclopedia). These qualities of a parabola would allow our company to provide HD quality TV shows, a more reliable service and a better experience than cable television. However, in order to fully understand my project, we need to first understand parabolas. The definition of a parabola is as follows: The set of all points in a plane that are equidistant from a fixed line, directrix, and [any fixed point] in the same plane or parallel plane (Dictionary.com). A parabola can be expressed through its quadratic or “general form.” Its quadratic form is expressed as: y = ax2+bx+c. Its general form is expressed as: (x-h)2 = 4p(y-k) if the directrix is parallel to the x-axis or (y-k)2 = 4p(x-h) if the directrix is parallel to the y-axis. However, we are more concerned with an equation that expresses the focus point of the parabola, because that is where the feed horn of a satellite is placed. A satellite has the same qualities of a parabola, a directrix, which is usually its mounting point, a vertex at the rounded point of the satellite, and a focus point, the feed horn.

(analyzemath.com) From the definition of the parabola, we know that any point (x,y) is equidistant from the focus and the directrix. The figure above demonstrates this quality. The figure shows a

parabola with a focus placed at (0, f) and a directrix at y = -f. To demonstrate the (x, y) is equidistant from the focus and the directrix, we can set up an equation as follows: √ √ Then we can simplify this equation further by squaring both sides and expanding the squares. Then we would get:

= = = = After performing basic math, we end up with the equation y = (x2/4f) with regard to the focus point. This equation is what will aid me in proving why a satellite works so well. Firstly, we know that a parabolic reflector, such as a satellite, transforms incoming waves into a spherical wave that converges to the focus or the feed horn on a satellite (Wikipedia: The Free Encyclopedia). This allows the feed horn on a satellite on a house to absorb all the electromagnetic energy and provide high quality TV shows. Also since the distance between satellites at home and the satellites in space is so large, this gathering of energy to a common focal point allows the satellites in peoples home to be able to attract energy from such a distant source. Previously, we derived the equation y = (x2/4f) to represent a parabola involving the focal distance f. From this equation it can be proved that what I said previously about a satellite being able to converge incoming waves towards the feed horn is true. To do this, we will need to use a calculus concept known as a derivative, see Appendix A to see how a derivative is found. The derivative of the aforementioned equation is y’ = x/2f. From the derivative we can perform several mathematic processes to prove that all incoming waves converge to the focus point. The figure below will be used and the math is as follows: (analyzemath.com)

The figure above shows a line, n, being reflected off a point on the parabola and being reflected towards the focus point F. The parabola can be seen as a satellite at a person’s home where line n is the incident ray from a satellite in space. The slope of the tangent line at point M(a, b) is represented as mt = a/2f, where a is substituted for the x in the derivative equation. The slope of mt and mn are related by -1 so it can be noted that mn = -2f/a. If n is the angle formed by the crossing of the incident ray at the x axis, then the relationship between this angle n and the slope of mn can be represented as tan(n) = -2f/a. Furthermore, it is noted that
Hence, the focus point is the y-intercept and we know that the focus point of a parabola is constant, so it can be deduced that the position of the incident ray does not matter, because all of the reflected rays will through the focus point of the satellite dish. This is why a feed horn is placed on a satellite dish. It absorbs all of the energy from the waves that hit the satellite. This proves that a satellite can focus energy at a given point, which can help our company provide our customers with high quality television shows. The following images show how a parabolic reflector receives and transmits waves. The images my statement that the feed horn of a satellite is receives all incoming energy waves. Receiving a Signal

Transmitting a Signal

(Oracle: Education Foundation, ThinkQuest)

With satellites being able to send distant signals, provide reliability, and have energy focused on one point, it will be in our company’s best interest to utilize them. By understanding parabolas, we can understand why satellites work so well. By providing our customers with satellite TV we can guarantee them a fantastic experience. I insist that we take the satellite approach if this company desires flourishment. Their parabolic qualities will greatly benefit this company.

Vertex

Directrix

Focus

Appendix A To find the derivative of a parabola represented by f(x) = x2/4f, we must first understand how finding the slope of two given points on a parabola works. Below, is a graph of a parabolic curve with point x. To find the derivative of the parabola, we must first find the slope of the tangent line at point x. However, before we do that, we must first understand how slope works on a shape, such as a parabola, where the slope is constantly changing between points. So, to begin, let’s find the slope of the secant line that crosses the coordinates (x, x2/4f) and (x+h, (x+h)2/4f) where x+h is a point h units away from x and (x+h)2/4f is simply f(x+h).

To find the slope we use the equation: where is the change in the ycoordinates and is the change in the x-coordinates. From the graph above, the change in y is f(x+h) – f(x) and the change in x is (x+h) – x. Therefore, we can represent the slope between the two points as:

We can also cancel out the x’s in the denominator because x-x=0. This leaves the denominator equal to h. Now we have the equation for the slope of a line that crosses through two points that are on the parabola. However, to find the derivative of the parabola, we need the slope of the tangent line that crosses through the point (x, f(x)). To do this we must analyze the graph. We know that h is any units away from x. As h gets closer and closer to x, then the slope of the secant line gets closer to the slope at x. So if we set a limit h 0, then we can find the slope at x. We are left with the equation:

When we evaluate the function for f(x) and f(x+h) we get the values (x2/4f) and ((x+h)2/4f). Next, we can plug in these values into the aforementioned equation, and use the new equation to find the derivative. We can also cancel out the x’s in the denominator since x – x = 0. We would then get the equation: (

)

When simplified using basic algebraic concepts, the equation equals to x/2f, which is the derivative of the equation f(x) = x2/4f hence f(x)’ = x/2f.

Work Cited “Concave Surfaces.” NEALS. N.p., Web. 5 Nov 2012 . "Find the Focus of Parabolic Dish Antennas.” Analyze Math. N.p., 21 2012 Web. 5 Nov 2012. . “How Parabolic Dish Antennas Work.” Analyze Math. N.p., 21 2012. Web. 5 Nov 2012. . "Mystery of Reflection." Oracle: ThinkQuest. N.p.. Web. 5 Nov 2012. . "Parabolic Reflector." Wikipedia: The Free Encyclopedia. N.p., 26 2012. Web. 5 Nov 2012. .