Mathematics Project

Student’s Name: Osama Zaid AL MENHALI 1011090131

Saeed Zaid AL MENHALI 1011090130

Abdullah Khamis AL AMERI 1011090072

Section: 12 -03

Term 2

2011 - 2012

2

Part I:

The figure below shows the graph of a parabola.

1. Plot the point A (3, 4.5), and draw the straight line () with equation y=−12

on the same graph above.

2. Calculate the distance between the points A and S (0, 0.5).

A (3 , 4.5) S (0 , 0.5)d= √ ( xA−xS )2+( yA− yS )2 = √ (3−0 )2+ (4.5−0.5 )2=5 3. Find the distance between the point A and the straight line ().

d= √ ( xA−x )2+ ( yB− y )2=√(3−3 )2+(4.5+0.5 )2=54. Compare the distances found in 2 and 3.

d2=d3

5=5

5. What is the name of the:

i. Point S: Focus

ii. Straight line (): Directrix

A (3 , 4.5)

3

Generalization:

Let P(x , y) is a point of the drawn parabola, and F (0 , p) the focus point of the parabola ( p>0 ) as illustrated in the figure below.

6. Find the distance d1 between P and F, and the distance d2 between P and the straight line with equation y=−p

d1= √ ( xF−xP )2+( yF− yP )2 = √ (0− x )2+( p− y )2=x+P− y d2= √ ( x−x )2+ ( y+ p )2= y+ p

7. Find the relation between x and y when d1 = d2.x+P− y= y+ px=2 y

4

Application

8. 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

y

x

i. Find an equation of the parabola.

V (0 , 0) latus rectum = | 1A|=10 a=

110

x=a ( y−k )2+h= 110

( y )2

ii. Find the diameter of the opening |CD|, 11 cm from the vertex.

x= 110

( y )2

11= 110

( y )2

y=±√110C (11 ,√110) D(11,−√110)

d= √ (11−11 )2+ (−√110−√110 )2=2√110cm

5

Part II

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 (s) before it received the signal from A.

(a) Assuming that radio signals travel at a speed of 980 ft/s, find an equation of the hyperbola on which the ship lies.

Lets FA (-200 , 0) & FB (200 , 0) so c = 200 mi

||AP|−|BP||=(1200 ×980 )=1176000 ft=245011

mi

||AP|−|BP||=2 a

2 a=245011

mi a=122511

mi

b2=c2−a2=40000−( 122511 )

2

=26598.1405

x2

12401.8595− y2

26598.1405=1

6

(b) If the ship is due north of B, how far of the coastline is the ship?

x2

12401.8595− y2

26598.1405=1

(200)2

12401.8595− y2

26598.1405=1

− y2

26598.1405=1− (200)2

12401.8595

− y2

26598.1405=−2.225322783

y=√(−26598.1405 ) (−2.225322783 )=243.2888161mi

7

Rubrics

Criteria Points

4 3 2 1

Completeness of Tasks

20%

Tasks are totally completed and correct. (100%)

Tasks are partially completed,

ORPartially wrong.

(75%)

Tasks are partially

completed,AND

Partially wrong (50%).

Tasks are Attempted (25%

or less)____

Presentation and Integration of Technology

70%

Students used one mean of technology. The tool used helped the student and was useful to support his project. Moreover, the student was able to explain the work he/she submitted confidently and fluently; he/she was able to answer all of colleagues and instructor’s questions

Student used a mean of technology but it was not that supportive to the topic. In addition, student was able to explain the work he/she submitted confidently and fluently and he/she reflected an understanding of his/her works. The student was able to answer most of colleagues and instructor’s questions.

Student was able to explain the work he/she submitted. Student reflected a shallow understanding of his/her work; she was able to answer some of colleagues and instructor’s questions,

Student use of technology was primitive and way below the level of other IAT students.Student was unable to explain the work he/she submitted. Student reflected no understanding of his/her work; he/she was unable to answer any of colleagues and instructor’s questions.

____

Creativity& enrichment

10%

Student had an outstanding addition in all aspects of his/her project.

Student had an outstanding addition in some aspects of his/her project.

Student had an outstanding addition in very few aspects of his/her project.

Student had an outstanding addition in no aspects of his/her project.

____

This rubric is out of 100, percentage orientation.

To make the mark out of 30 (Student’s Mark/10*3)Total ----> ____