Developing Expert Voices

Developing Expert Voices(D.E.V)

Presented By:Alanna Lam & Linh Trinh

Problem #1

Prove This Identity:

Trigonometric Identities

cos2x = 1 - tan²x1 + tan²x

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

1cos²x

* Remember tan can also be written as sin/ cos.

* Remember sec is the inverse of cos.

Solution (cont’d):

1 - tan²x sec²x

cos2x = 1 - tan²x 1 + tan²x

1 - sin²x cos²x

1cos²x

cos²x - sin²xcos²x 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

Solution (cont’d):

1 - tan²x sec²x

cos2x = 1 - tan²x 1 + tan²x

1 - sin²x cos²x1cos²x

cos²x - sin²xcos²x cos²x( ) cos²x

1

cos²x -cos²x

( )sin²x cos²x 1

cos²x - sin²x

cos(x + x)

cosxcosx - sinxsinx

cos²x - sin²x

Q. E. D

Problem #2Combinatorics

There are 7 couples seated around a circular table.

a)How many ways can they seat themselves randomly?

b)How many can they seat themselves if the couples insist on sitting together?

c)How many ways are there is the men and women alternate?

Solution:a)How many ways can they seat

themselves randomly?

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

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

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

Problem #3Logarithms

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.

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 000e0.023y

P= 238 000(1.0232)y

e0.017 = 1.0232

Formula: Ao(Model)t

Solution (cont’d)

b) Calculate how many years it would take for the population to double.

Po = 238 000P = 2Po = 476 000

476 000 = 238 000e0.023y238 000 238000

2 = e0.023y

ln2 = lne0.023y

ln2 = 0.023y

ln2 = y0.0

2330.1368 = y

Approximately during the 30th year the population in Hanoi will double.

Isolate the y

Solution (cont’d)

c) Calculate when the population will reach 1 million.1 000 000 = 230 000e0.023y

1 000 000 = e0.023y230 000

ln 1 000 000 = 0.023y238

000( )1 ln 1 000 000 = y0.023

230 000( )

1 ( 1.4355) = y0.0

23y = 62.4130

In approximately 62 years the population would reach 1 million.

Problem #4Probability

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.

a)Draw a sample space of all possible outcomes using a tree diagram.

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?

Solution:a)Draw a sample space of all possible

outcomes using a tree diagram.

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% ( () )1

118 19

8

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% ( () )1

118 19

8

Problem #5Conics

For the eclipse whose equation is given below:x2 + 4y2 =

16a)Rewrite this equation in standard

form.

b)Sketch a graph.

c)Find the major and minor axes, coordinates of the vertices and foci.

Solution:

a) Rewrite this equation in standard form.

x2 + 4y2 = 16

x2 + 4y2 = 16

x2 + y2 = 116 4

a)Sketch a graph.

cF2 F1

Solution (cont’d)

a)Find the major and minor axes, coordinates of the vertices and foci.

Major Axis: 2aMinor Axis: 2b

c

Length of c:

c2 = b2 - a2

c2 = 16 – 4

c2 = 12

c = √12

= 82(4)

= 42(2)

Verticies:

A1 (4, 0)

A2 (-4, 0)

B1 (0, 2)

B2 (0,-2)

Foci:

F1 (√12, 0)

F2 (-√12, 0)

F1F2

Alanna’s Reflection:• Why did you choose the concepts you did to create

your problem set?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.

• 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.

• Did you learn anything from this assignment? Was it educationally valuable to you?

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.

Thank You for viewing our D.E.V project

The End