The document presents solutions to 5 math problems to demonstrate mathematical understanding. Problem 1 solves a trigonometric identity. Problem 2 uses combinations and permutations to solve seating arrangements. Problem 3 involves exponential growth and logarithms. Problem 4 uses probability and tree diagrams. Problem 5 analyzes a conic section using standard form equations. The creator reflects that working through challenging problems with a partner helped improve their conceptual understanding.

1. Developing Expert Voices (D.E.V) Presented By: Alanna Lam & Linh Trinh

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. 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. 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

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. 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. 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. 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. 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

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

21. T hank You for viewing our D.E.V project The End