Thursday, February 6, 2014

RWA#1: Unit M Concept 5: Graphing ellipses given equation and identifying all parts(center,focus,major and minor axis, vertices and covertices,and eccentricity)

1. "The set of all points such that the sum of the distance of two points is a constant."
2. An ellipse in standard form will have the equation that looks like (x-h)^2/a^2+(y-k)^2/b^2=1 or (x-h)^2/b^2+(y-k)^2/a^2. If the bigger number is under the x, then the ellipse is going to be fat. If the bigger number is under the y, then is going to be skinny. One way to remember is "y so skinny" and "you are x-tra large." An ellipse looks like a squished out/up circle. An ellipse has all of the following:center,focus,major and minor axis, vertices and covertices,and eccentricity. If you are given the general equation of an ellipse you are going to have to complete the square(it has to equal one) in order to get the equation into standard form. Once, you have it into standard form you can right away get the center. Remember that x goes if h and y goes with k and the signs will switch. Now looking at the denominator of the equation you have to see if the bigger number is under the x, then the ellipse is going to be fat and if the bigger number is under the y, then it is going to be skinny. Again looking back at the denominator of the equation, the bigger number is always "a" but you need to take the square root of that number. The other number is "b" and you also take the square root of that number. In order to find "c" you will need to plug in the two variables that you know into the formula a^2-b^2=c^2 and solve for c. To find the vertices you will have to look at your center and whether your graph is going to be fat or skinny. If your graph is fat then then you add and subtract "a"to x of the center and y will stay the same and vice versa goes if your graph will be skinny. Therefore, your major axis will be y= the y of the center(the major axis has the vertices,center, and focus)and vice versa goes if your graph will be skinny. To find the co vertices you will have to look at your center and whether your graph is going to be fat or skinny. If your graph is fat then then you add and subtract "b" to y of the center and x will stay the same and vice versa goes if your graph will be skinny. Therefore, your minor axis will be x= the x of the center and vice versa goes if your graph will be skinny. Major axis are solid lines and minor axis are dotted lines. To find the eccentricity you plug in "c" and "a" into the formula:e=c/a. In order to find the foci you have to look at your center and whether your graph is going to be fat or skinny. If your graph is fat then then you add and subtract "c"to the x of the center and y will stay the same and vice versa goes if your graph will be skinny.An ellipse has to have an eccentricity of greater than zero but less than one. Eccentricity is just "a measure of how much the conic section deviates from being circular."  "As the distance between foci increases the eccentricity increases, or the reverse relationship."
To get more information on how to find the parts of an ellipse watch below.







3. A real world application of an ellipse is an extracorporeal shockwave lithotripsy, which "enables doctors to treat kidney and gall stones without open surgery"(http://www.lupdirect.com/urologicalprocedures_lithotripsy.php)
Basically,external, high-intensity, and focused shock waves  pass through the body and they reach the  stones and the pressure causes the stone to be stressed. Eventually, they  fracture into smaller fragments that can be pass by the patient easy. Ultrasound or fluoroscopic x-ray system is used to direct the focus of the waves accurately on the stone. This is better than a doctor performing open surgery, which doctors can damage surrounding tissue and the risk of infections.
The extracorporeal shockwave lithotripsy has half of a  three dimensional  representation of an ellipse piece that is sitting on the patient's side. The lithotripter works because of the reflectiveness of an ellipse.
4.Works Cited
  • http://www.lessonpaths.com/learn/i/unit-m-conic-sections-in-real-life/conic-sections-in-real-life
  • http://www.lupdirect.com/urologicalprocedures_lithotripsy.php
  • http://www.sophia.org/tutorials/unit-m-concept-5a?cid=embedplaylist

No comments:

Post a Comment