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Math 110/003 - Group 7
Members:	Byron Godoy Pinzon
	Michelle Little
	Emily Oates
	Jose Torres-Torija Cubillas

Homework 4

Question #1

Ok, so first we assigned the cat to Tosh, the frog to Bianca, the parrot to Jaela, and the snake to Jun. Then, we knew that the cat was named Jun and the frog was named Suzan. The owner of the turtle was the name of the owner of Tosh. So, we discarded Jun (because she was already named after the cat) and Suzan (because she was the name of the frog). That left us with Jaela. So, that means that Tosh is the parrot (because Jaela owns the parrot) and the turtle's name is Jaela. So that left us with Jun naming the snake after Bianca. And we know that Suzan's mother's name is Jun because the question states that her mother's pet's name is Bianca.

In conclusion: Tosh owns a cat (named Jun) Bianca owns a frog (named Suzan) Jaela owns a parrot (named Tosh) Suzan owns a turtle (named Jaela) Jun owns a snake (named Bianca)

Question #2

Bohao, Stewart, Dylan, Tim and Chan are the five players of a basketball team. Two are left handed and three right handed, Two are over 2m tall and three are under 2m, Bohao and Dylan are of the same handedness, whereas Tim and Chan use different hands. Stewart and Chan are of the same height range, while Dylan and Tim are in different height ranges. If you know that the one playing centre is over 2m tall and is left handed, can you guess his name?

Tim

So, this problem is solved by writing out the information: 2 people are left handed, 3 are right handed, 2 are over 2m and 3 are under 2m; Bohao and Dylan = same hand, Tim =/= Chan in hand; Stewart and Chan = same height, Dylan =/= Tim (height) since Bohao and Dylan have the same hand and Tim and Chan don't, that means that Bohao and Dylan and ONE OF Tim or Chan are right handed. since Stewart and Chan have the same height and Dylan and Tim don't, that means that Stewart and Chan and ONE OF Tim and Dylan are under 2m tall.

so when correlating this information with each player, you get:

Bahao: right handed, over 2 m tall Stewart: left handed, under 2m tall Dylan: right handed, either over / under 2m tall Tim: either right / left handed, either over / under 2m tall Chan: either right / left handed, under 2m tall

If the one playing centre is OVER 2m tall and is LEFT handed that takes out Bahao, Stewart, Dylan and Chan by eliminating the factors, so the centre must be Tim.

Question #3

Determine the positions of each player on the baseball team. Outfielders:

           center fielder -- Sung
           left fielder -- Ed
           right fielder -- Bobo

Infielders:

           1st baseman -- Pascal
           2nd baseman -- Charles
           3rd baseman -- Adam
           Shortstop -- Jason

Battery:

           Pitcher -- Hassan
           Catcher -- Mathieu

We determined this by process of elimination and by using the clues to determine what position the players for sure were NOT.

-Clue #1: Adam does not like the catcher --> hence, Adam is NOT the catcher

-Clue #2: Ed's sister is engaged to the second baseman --> the 2nd baseman is NOT single

-Clue #3: The centre fielder is taller than the right fielder --> upon figuring out with later clues that Bobo was the right fielder, it made sense that Sung was center fielder

-Clue #4: Hassan and the third baseman live in the same building --> Hassan is NOT the third baseman

-Clue #5: Pascal and Charles each won $20 from the pitcher at a poker game --> Neither Pascal or Charles is the pitcher

-Clue #6: Ed and the outfielders play cards during their free time --> thus, Ed is NOT an outfielder

-Clue #7: The pitcher's wife is the third baseman's sister --> the pitcher is married

-Clue #8: All the battery and infield except Charles, Hassan and Adam are shorter than Sung --> Thus, Charles, Hassan, and Adam cannot be in the outfield

-Clue #9: Pascal, Adam and the shortstop lost $100 each at the race track -->Pascal and Adam cannot be the shortstop

-Clue #10: The second baseman beat Pascal, Hassan, Bobo and the catcher at billiards -->Pascal, Hassan nor Bobo can be the catcher OR the 2nd baseman

-Clue #11: Sung is in the process of getting a divorce --> Thus, Sung is SINGLE

-Clue #12: The catcher and the third baseman each have two legitimate children --> the catcher and the 3rd basemen are married

-Clue #13: Ed, Pascal Jason, the right fielder and the centre fielder are bachelors, the others are all married --> Ed, Pascal, Jason, the right fielder and the center fielder must not be married

-Clue #14: The shortstop, the third baseman and Bobo all attended the fight --> Bobo cannot be the shortstop or the 3rd baseman

-Clue #15: Mathieu is the shortest player of the team --> he cannot be outfield

Through the process of elimination, we determined the results shown above. We concluded the following facts about each player: Adam: could NOT be the catcher, 2nd baseman, right, left, or center outfielder or the shortstop Bobo: could NOT be the shortstop, 3rd baseman, the catcher, or 2nd baseman Charles: could NOT be the pitcher, right, left, or center outfielder. Ed: could NOT be right fielder, center fielder, catcher, 3rd baseman, pitcher or 2nd baseman Hassan: could NOT be 2nd baseman, catcher, right, left or center outfielder, or 3rd baseman Jason: could NOT be right fielder, center fielder, 3red baseman, catcher, or pitcher Mathieu: could NOT be right, left or center outfielder Pascal: could NOT be pitcher, 2nd baseman, catcher, right fielder, center fielder, 3rd baseman, or shortstop Sung: single, and tall

From these conclusions about each person, we put each possible candidate next to each position, and by crossing off each name we had found the position for, we solved the puzzle:

Right fielder: Bobo Left fielder: Ed, Jason, Pascal, Sung, Bobo Center fielder: Bobo, Sung 1st baseman: Adam, Bobo, Charles, Hassan, Jason, Mathieu, Pascal, Sung 2nd baseman: Charles, Ed, Jason, Mathieu, Sung 3rd baseman: Adam, Charles, Mathieu, Sung Shortstop: Charles, Hassan, Jason, Mathieu, Sung Pitcher: Adam, Bobo, Hassan, Mathieu Catcher: Charles, Mathieu, Sung

Question #4

Since you do not know the actual line up of the tournament you must create scenarios were every competitor must play a game versus each other only once. By doing this we can create random matches with players we know are not the “main” attraction and eventually find out who Fernanda will play on the fifth day.

-On the first day, we know that Carla played Petra then that means Fernanda could have played against Sandra, Janet, or Li. So we can set up matches as is Carla vs. Petra, Fernanda vs. Li, Janet vs. Sandra C.vs.P F.vs.L J.vs.S

-On the second day, we know that Carla played Janet then that means Fernanda could have played Petra, Sandra, or Li. Since on the first day Fernanda and Li already played each other they cannot play each other again so we must put her with Sandra. So the matches are Carla vs. Janet, Fernanda vs. Sandra, and Petra vs. Li C.vs.J F.vs.S P.vs.L

-On the third day, If Janet played Li then that means Fernanda could have played Petra, Sandra or Carla. Since we know that Fernanda has already played against Sandra we cannot match her up with her so she must play against Petra. So matches are Janet vs. Li, Fernanda vs. Petra, Carla vs. Sandra J.vs.L F.vs.P C.vs.S

-On the fourth day, If Petra played Sandra then that means Fernanda could have played Carla, Janet or Li. But sense Fernanda already played against Petra and Li she must play Janet because otherwise Carla would play against the same person. Petra vs. Sandra, Fernanda vs. Janet, Carla vs. Li P.vs.S F.vs.J C.vs.L

-On the firth day since each player only plays each of the others once, the only matches left of are those that involve people not playing each other before hand so the matches are Petra vs. Janet, Li vs. Sandra and Fernanda vs. Carla. P.vs.J F.vs.C L.vs.S

So on the 5th day Fernanda must play against Sandra

Question #5

Homer finally had a week off from his job at the nuclear power plant and intended to spend all nine days of his vacation (Saturday through the following Sunday) sleeping late. But his plans were foiled by some of the people who work in his neighbourhood. On Saturday, his first morning off, Homer was wakened by the doorbell; it was a salesman of magazine subscriptions. On Sunday, the barking of the neighbour's dog abruptly ended Homer's sleep. On Monday, he was again wakened by the persistent salesman but was able to fall asleep again, only to be disturbed by the construction workers next door. In fact, the salesman, the neighbour's dog and the construction workers combined to wake Homer at least once each day of his vacation, with only one exception. The salesman woke him again on Wednesday; the construction workers on the second Saturday; the dog on Wednesday and on the final Sunday. No one of the three noisemakers was quiet for three consecutive days; but yet, no pair of them made noise on more than one day during Homer's vacation. On which day of his holiday was Homer actually able to sleep late?

Salesman = S Dog = D Construction Worker = CW Homer had a vacation from Saturday through to the next Sunday

Saturday -> S

Sunday -> D

Monday -> S+CW

Tuesday ->

Wednesday -> S+D

Thursday -> CW+D

Friday ->

Saturday -> CW+S

Sunday -> D

Homer would get sleep on Tuesday and Friday during his vacation. This will work if every noise maker made noise on the 3rd day of the non-consecutive day period. The Salesmen would have to show up the first Saturday, Monday, Wednesday and then the second Saturday. The dog then makes noise on Sunday, Wednesday, Thursday and Sunday the dog makes noise on 2 consecutive days so that he can pair up with the construction worker. So then the construction worker makes noise on Monday, Thursday and Saturday.

Homework 5

Those That Pose No Problem

Basic functions, Basic Properties of Functions, Equations, Polynomial Long Division, Graphs of Functions, Intersections of Functions, Reading Graphs of functions, Distances and Lines, Media:Example.ogg Operations on Graphs of Functions, Construction of Graphs, Trigonometry and the Pythagoras Theorem, Areas and Volumes, Mathematical Writing.

Those That Just Need Practice

Inequalities and Composition of Functions

Basic Skills Learning Guide Project

Learning Outcomes With Regards To Equations

Example of Solving Linear Equations

Definition of a linear equation: A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable.

Example 1:

3/4x +5/6 = 5x - 125/3

Multiply both sides by the lowest common multiple of 4, 6, and 3, or 12.

12(3/4x +5/6) = 12(5x -125/3)

Simplify:

12(3/4x) + 12(5/6) = 12(5x) - 12(125/3)

3(3x) + 2(5) = 60x - 500

9x + 10 = 60x - 500

Subtract 9x from both sides of the equation:

10 = 51x - 500

Add 500 to both sides of the equation:

510 = 51x

Divide both sides by 51:

x = 510/51 = 10

Example of Solving Quadratic Equation

Definition of a quadratic equation: a quadratic equation is a polynomial equation of the second degree.

Example 1:

Solve 6x^2 + 11x – 35 = 0.

This may factor, but it looks like it would be a fair amount of work, and I'm feeling a bit mindless and lazy at the moment, so I'll use the Quadratic Formula instead:

-b + or - sqrt(b^2 - 4ac)/2a

x = -(11) + or - sqrt(11)^2 - (4)(6)(-35)/(2)(6)

x = -11 + or - sqrt(121 + 840)/12

x = -11 + or - sqrt961/12

x = -11 + or - 31/12

x = -11 - 31 / 12, -11 + -31/12

x = -42/12, 20/12

x = -7/2, 5/3

The solution is x = –7/2, 5/3

Example of Solving Linear Inequalities

Definition of a linear inequality: an inequality which involves a linear function.

Example 1:

Solve (2x – 3)/4 < 2.

First, I'll multiply through by 4. Since the "4" is positive, I don't have to flip the inequality sign:

(2x – 3)/4 < 2

(4) × (2x – 3)/4 < (4)(2)

2x – 3 < 8

2x < 11

x < 11/2 = 5.5

Example of Solving Quadratic Inequalities

Definition of a quadratic inequality: an inequality in which one side is a quadratic polynomial and the other side is zero.

Example 1:

Solve x^2 + 2x – 8 < 0. First, find the zeroes:

x^2 + 2x – 8 = 0

(x + 4)(x – 2) = 0

x = –4 or x = 2

These x-intercepts split the number line into three intervals: x < –4, –4 < x < 2, and x > 2. Since this is a "less than" inequality, I need the intervals where the parabola is below the x-axis. Since the graph of y = x2 + 2x – 8 is a right-side-up parabola, it is below the axis in the middle (between the two intercepts). Since this is an "or equal to" inequality, the boundary points of the intervals (the intercepts themselves) are included in the solution.

Then the solution is: –4 < x < 2

A website which we found extremely helpful in understanding equations can be found here:

http://www.sosmath.com/algebra/solve/solve0/solve0.html

This website contains questions involving linear equations, equations containing radicals, equations containing absolute values, quadratic equations, equations involving fractions, exponential equations, logarithmic equations and trigonometric equations. In addition to all these questions, it provides you with the solution in a step by step manner, which are very easy to understand.

In addition, there are two youtube videos which also helped contribute to our understanding. They can be found at these links:

The first video is about solving rational equations, while the second is about linear equations.

Group Project: Basic Skills and Equations

What is an equation?

An equation is an expression or proposition, often algebraic, asserting the equality of two quantities

What is a linear equation?

A linear equation is a mathematical expression that has an equal sign and linear expressions. A variable is a number that you don't know, often represented by "x" or "y".

To solve a linear equation -- you simply want to move all other numbers and coefficients to isolate ONE variable. So, for instance, if you wanted to solve 2x-11=3 then:

Step 1: start by writing out the original equation

2x-11=3

Step 2: begin isolating x by adding 11 to both sides

2x=14

Step 3: isolated x completely by dividing both sides by 2

x=7

You can check this by adding in your new value for x into the original equation which will give you 3.

Another example: solve for x when 4x-4y=8

Step 1: write out the original equation

4x-4y=8

Step 2: add 4y to both sides to isolate one variable on one side

4x=8+4y

Step 3: isolate x by dividing both sides by 4

x=2+y

For a video on how to solve linear equations, please see

http://www.youtube.com/watch?v=ldYGiXSHa_Q

Or, for a video on how to solve linear equations involving inequalities, please see:

http://www.youtube.com/watch?v=0X-bMeIN53I&feature=channel

What is a quadratic equation?

A quadratic equation is a polynomial equation of the second degree, typically looking like

ax^2+bx+c=0

Where x represents a variable and a, b, and c are constants where a cannot equal 0

If a = 0, then this would be a linear equation

In general, quadratic equations must have an x^2 term and NO higher

Quadratic equations can be solved by factoring, completing the square, graphing, Newton’s method, and using the Quadratic Formula.

The quadratic formula looks like x=(-b±√(b^2-4ac))/2a

Where the symbol ± means that both x=(-b-√(b^2-4ac))/2a and x=(-b+√(b^2-4ac))/2a are solutions

Some examples of how to solve it:

Solving by Factoring:

x^2+2x-15=0

Step 1: factorize the quadratic equation:

(x-5)(x+3)=0

Step 2: set each factor equal to 0:

(x-5)=0 or (x+3)=0

Step 3: solve the resulting linear equations:

x=5 or x=-3

To check: plug the solutions in the original equation:

Solving by Completing the Square:

Find the roots of x^2 + 10x − 4 = 0

Step 1: a = 1 [no action necessary]

Step 2: Rewrite the equation with the constant term on the right side.

x^2 + 10x = 4

Step 3: Complete the square by adding the square of one-half of the coefficient of x to both sides. In this case:

〖(10/2)〗^2= 〖(5)〗^2=25

x^2 + 10x + 25 = 4 + 25

x^2 + 10x + 25 = 29

Step 4: Write the left side as a square:

(x + 5)^2 = 29

Step 5: Equate and solve

x+5= ± √29

x=-5 ± √29

Solving by using the Quadratic Formula:

By inspection, we can see that: a = 2, b = -7 and c = -5

Substituting these into the quadratic formula, we get:

x=(-b±√(b^2-4ac))/2a

x=(-(-7)±√(〖(-7)〗^2-4(2)(-5)))/(2(2))

x=(7±√(49+40))/4

x=(7±√89)/4

x=(7+√89)/4 or x=(7-√89)/4

For a video on how to solve quadratic equations, please see

http://www.youtube.com/watch?v=9ETZbifO0fM&NR=1

What are polynomial equations of degrees more than 3?

A cubic equation is an equation involving a cubic polynomial, i.e., one of the form

a_3 x^3+a_2 x^2+a_1 x+a_0=0

Where a_3 cannot be 0 or else it would be a quadratic equation

To watch a video on how to solve cubic equations using the factor theorem, please see

http://www.youtube.com/watch?v=FBFHBxnAU9I

For an overall review on how to solve cubic equations, please see

http://www.youtube.com/watch?v=M-xghh_tvDs&feature=related

A quartic equation is a fourth-order polynomial equation of the form

x^4+a_3 x^3+a_2 x^2+a_1 x+a_0=0

What are equations of square roots?

The main idea behind solving equations containing square roots is to raise to power 2 in order to clear the square root using the property

〖(√x)〗^2=x

The above holds only for x greater than or equal to 0. In solving equations we square both sides of the equations and instead of putting similar conditions we check the solutions obtained.

Example: Find all real solutions to the equation √(x+1)=4

√(x+1)=4

Step 1: raise both sides to the power 2 to clear the square root:

(√(x+1) )^2=4^2

Step 2: simplify:

x+1=16

Step 3: solve for x:

x=15

To check, plug this value for x into the original equation.

Another Example: Find all real solutions to the equation √(3x+1)=x-3

√(3x+1)=x-3

Step 1: raise both sides to the power 2 to clear the square root:

(√(3x+1) )^2=(𝑥x−3)^2

Step 2: simplify:

3x+1=x^2-6x+9

Step 3: write the equation with right side equation to 0:

x^2-9x+8=0

Step 4: it is now a quadratic equation with 2 solutions:

(x-8)(x-1)=0

x=8 and x=1

For a video involving equations with square roots, please see:

http://www.youtube.com/watch?v=V8I2_zgNRHI

For more information on solving different types of equations, you can visit the sites:

http://www.sosmath.com/algebra/solve/solve0/solve0.html

http://www.analyzemath.com/equations_inequalities.html

Or visit youtube.com and enter the type of equation you would like solved and free tutorials will give you answers.