Developing Expert Voices

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Developing Expert Voices

  1. 1. Developing Expert Voices (D.E.V) Presented By: Alanna Lam & Linh Trinh
  2. 2. Problem #1 <ul><li>Prove This Identity: </li></ul>Trigonometric Identities cos2x = 1 - tan²x 1 + tan²x
  3. 3. Solution: A good first step is to draw a line to divide the equation at the equal sign 1 - tan²x sec²x * Remember 1 + tan²x can also be written as sec²x. cos2x = 1 - tan²x 1 + tan²x 1 - sin²x cos²x 1 cos²x * Remember tan can also be written as sin/ cos. * Remember sec is the inverse of cos.
  4. 4. Solution (cont’d) : 1 - tan²x sec²x cos2x = 1 - tan²x 1 + tan²x 1 - sin²x cos²x 1 cos²x cos²x - sin²x cos²x cos²x ( ) cos²x 1 * Remember the # 1 can be written in many ways. cos/ cos is the same thins as 1 Multiplying by the reciprocal cos²x - cos²x ( ) sin²x cos²x 1 cos²x - sin²x
  5. 5. Solution (cont’d) : 1 - tan²x sec²x cos2x = 1 - tan²x 1 + tan²x 1 - sin²x cos²x 1 cos²x cos²x - sin²x cos²x cos²x ( ) cos²x 1 cos²x - cos²x ( ) sin²x cos²x 1 cos²x - sin²x cos(x + x) cosxcosx - sinxsinx cos²x - sin²x Q. E. D
  6. 6. Problem #2 Combinatorics <ul><li>There are 7 couples seated around a circular table. </li></ul><ul><li>How many ways can they seat themselves randomly? </li></ul><ul><li>How many can they seat themselves if the couples insist on sitting together? </li></ul><ul><li>How many ways are there is the men and women alternate? </li></ul>
  7. 7. Solution: <ul><li>How many ways can they seat themselves randomly? </li></ul>Good to start off with a reference point. So let us start off by seating one person as the reference point. Formula: (n -1)! # of people First seated (14 – 1)! = 6227020800 ways
  8. 8. Solution (cont’d) b) How many can they seat themselves if the couples insist on sitting together? 7 6 5 4 3 2 2! 2 ! 2 ! 2 ! 2 ! 2 ! 2 ! 2! A helpful tip is to put the couples in a “bag” and once arranged you can rearrange the couples in the bag. =645120
  9. 9. Solution (cont’d) c) How many ways are there if the men and women alternate? Lets seat the ladies first. Then once they are seated we will alternate the men. Ladies x Men (7 – 1)! x 7! 6! x 7! = 3628800 ways
  10. 10. Problem #3 Logarithms In 1950 the population in Hanoi was 238 000 and is increasing at the rate of 1.7% per year. a) Write an equation to represent the population of Hanoi, as a function of the number of years, “y”, since 1950. b) Calculate how many years it would take for the population to double. c) Calculate when the population will reach 1 million.
  11. 11. Solution: a) Write an equation to represent the population of Hanoi, “H”, as a function of the number of years, “y”, since 1950. P= 238 000e 0.023y P= 238 000(1.0232) y e 0.017 = 1.0232 Formula: A o (Model) t
  12. 12. Solution (cont’d) b) Calculate how many years it would take for the population to double. P o = 238 000 P = 2P o = 476 000 476 000 = 238 000e 0.023y 238 000 238000 2 = e 0.023y ln2 = lne 0.023y ln2 = 0.023y ln2 = y 0.023 30.1368 = y Approximately during the 30 th year the population in Hanoi will double. Isolate the y
  13. 13. Solution (cont’d) c) Calculate when the population will reach 1 million. 1 000 000 = 230 000e 0.023y 1 000 000 = e 0.023y 230 000 ln 1 000 000 = 0.023y 238 000 ( ) 1 ln 1 000 000 = y 0.023 230 000 ( ) <ul><li>( 1.4355) = y </li></ul>0.023 y = 62.4130 In approximately 62 years the population would reach 1 million.
  14. 14. Problem #4 Probability <ul><li>Hank decides to go to Toronto for a job interview. The probability that he will MOVE in at Toronto is 6/11. Probability that he will move to Toronto and get the job is 10/18. The probability of him getting the job but NOT moving to Toronto is 2/18. </li></ul><ul><li>Draw a sample space of all possible outcomes using a tree diagram. </li></ul>c) What is the probability that he doesn’t get the job and doesn’t move into Toronto? b) What is the probability that he will get the job and move into Toronto?
  15. 15. Solution: <ul><li>Draw a sample space of all possible outcomes using a tree diagram. </li></ul>
  16. 16. Solution: b) What is the probability that he will get the job and move into Toronto? P (M,GJ) = 6 10 = 60 = 0.3030 = 30% ( ( ) ) 11 18 198 c) What is the probability that he doesn’t get the job and does move into Toronto? P (M, NJ) = 6 8 = 42 = 0.2121 = 21% ( ( ) ) 11 18 198
  17. 17. Problem #5 Conics For the eclipse whose equation is given below: x 2 + 4y 2 = 16 <ul><li>Rewrite this equation in standard form. </li></ul><ul><li>Sketch a graph. </li></ul><ul><li>Find the major and minor axes, coordinates of the vertices and foci. </li></ul>
  18. 18. Solution: a) Rewrite this equation in standard form. x 2 + 4y 2 = 16 x 2 + 4y 2 = 16 x 2 + y 2 = 1 16 4 <ul><li>Sketch a graph. </li></ul>c F2 F1
  19. 19. Solution (cont’d) <ul><li>Find the major and minor axes, coordinates of the vertices and foci. </li></ul>Major Axis: 2a Minor Axis: 2b c Length of c: c 2 = b 2 - a 2 c 2 = 16 – 4 c 2 = 12 c = √12 = 8 2(4) = 4 2(2) Verticies: A1 (4, 0) A2 (-4, 0) B1 (0, 2) B2 (0,-2) Foci: F1 (√12, 0) F2 (-√12, 0) F1 F2
  20. 20. Alanna’s Reflection: <ul><li>• Why did you choose the concepts you did to create your problem set? </li></ul><ul><li>Me and my partner thought we both had challenges on the units we chose. So we made up questions based on what we thought was challenging and we succeeded. </li></ul><ul><li>• How do these problems provide an overview of your best mathematical understanding of what you have learned so far? We chose questions that made sure we had difficulty with. We had both worked together on finding the correct answer which help us understand the concept. </li></ul><ul><li>• Did you learn anything from this assignment? Was it educationally valuable to you? </li></ul><ul><li>YES! This project cleared my mind a little more than before. Over all I am quite glad we did this although it was so time consuming. </li></ul>
  21. 21. T hank You for viewing our D.E.V project The End

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