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Math Project

Math Project . Done By: Mubarak Ali Grade : 12.02. Goals. Students will interpret their understanding of applications of conic sections Students will use technology for research and communication: wiki/blog.

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Math Project

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  1. Math Project Done By: Mubarak Ali Grade : 12.02

  2. Goals Students will interpret their understanding of applications of conic sections Students will use technology for research and communication: wiki/blog. Students will present their findings to the class through digital Media ( Video/Booklet).

  3. Theme • This project covers the application of topics in conic sections; students will • apply the concepts and theorems they have learned on circles, parabolas, ellipses and hyperbolas. • Moreover students will be able to apply their knowledge in relation to various Engineering problems.

  4. Task 1 i) Find at least three real life application pictures representing the parabola.

  5. Task 1 ii) To hear what athletes are saying on the field or on the course, sports reporters use microphones with parabolic shields to concentrate the sound at a microphone. For optimum audio reception, the microphone should be placed at the focus of this parabola. Determine the coordinates of the focus, then write the equation of the parabola. X=a (y-k) 2+h X=a (y) 2 0.75=a (2.5) 2 0.75/(2.5) 2 = a A=0.12 The equation is: X=0.12y2 Determine the focus: V (0,0) F (0+1/4a, 0) F (0+1/4a(0.12), 0) =(25/12 , 0)

  6. Task 2 • Part 2:
Solar System: • The elliptical orbit of Pluto has the greatest eccentricity among all planets of our solar system (𝑒 = 0.248). Pluto’s orbit has the sun at one focus and a major axis of 79.6 AU (astronomical units). • Consider Pluto’s orbit on a giant coordinate grid where x and y are measured in AU and the sun is at the origin. (As shown in the figure above)

  7. Part 2 A.Usethe length of the major axis to determine a. 2a=79.6 a=39.8 B. Use the eccentricity and 𝑎 to determine the focal radius 𝑐. (Round to 2 decimal places.) E=c/a 0.248=c/39.8 c=9.87 C. Write an equation of the form (𝑋−𝐶)2 + 𝑦2 = 1 to model Pluto’s orbit. 𝑎2 𝑏2 A2=b2+c2 39.8=b2+(9.87)2 b=38.56 b2=(39.8)2-(9.87)2 b2=1486.62

  8. Part 2 • D . The perigee is when the Pluto is closest to the sun. The apogee is when it is • furthest from the sun. What are the distances from the • sun at these points?

  9. Part 3 Part 3: Bridge: Mohamed and Salim study a drawing of an ornamental bridge such as the one shown at the right. It shows an elliptical arch that spans a narrow strait of water. The arch they are studying can be modeled by the following formula 𝑥2 𝑦2 182.25 + 132.25 = 1 Mohamed wants to know the dimensions of the arch A.Whichterm in the equation defines the horizontal axis? a2=182.25 a=13.5 B. Write a comparative statement to show whether the major axis of the arch 
is horizontal or vertical a2 > b2=horizontal C. Mohamed says that the width of the bridge is 13.5 feet. Is he correct? 
Explain. B=square root 132.25 = 11.5 D. Write an expression for the height of the bridge. B=11.5

  10. Part 3 Salim notes that this bridge is hardly high enough to pass under while standing up in a moderate-size boat. If he were to build a bridge, it would be at least 1.3 times as wide and twice as high. Find the vertices and co-vertices of Salim’s bridge design. 2b(1.3)=46 2a(2)=54 Write an equation for the design of Salim’s bridge using his minimum 
dimensions 2b=46 > b=23 2a=54 > a=27 X2/272+Y2/232=1 X2/729+Y2/529=1

  11. Part 4 Task 4 A cross-section of a parabolic reflector is shown in the figure.
The bulb is located at the focus and the opening at the focus is 10 cm Find an equation of the parabola. Find the diameter of the opening |CD|, 11 cm from the vertex. 1. X=ay2 (P, 5) P= a(5)2 = p=25a p=1/4a 1/4a=25a 100a2=1 a2=1/100 A=square root 1/100 =1/10 2. X=1/10 y2 11=1/10y2 =y2 =110 y=+-square root 110 CD= 2 square root 110

  12. Part 5 Part 5 In the LORAN (Long Range Navigation) radio navigation system, two radio stations located at A and B transmit simultaneous signals to a ship located at P. The onboard computer converts the time difference in receiving these signals into a distance difference |PA| - |PB|, and this, according to the definition of a hyperbola, locates the ship on one branch of a hyperbola (see the figure). Suppose that station B is located 400 mi due east of station A on a coastline.
A ship received the signal from B 1200 microseconds (ms) before it received the signal from A. Assuming that radio signals travel at a speed of 980 ft/ms, find an equation of the hyperbola on which the ship lies. PA-PB=2a 2a=980X1200 1176000ft = 1176000 x 0.000189393939 2a=222.7 a=111.35 c2=a2+b2 b2=c2-a2 =(200)2-(111.35)2 = 166.14 (b) If the ship is due north of B, how far of the coastline is the ship? 2002111.352-Y2/166.142=1 y=247.8 mile

  13. The End

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