SAMUS'

Developing Expert Voices

2007-04-25T22:31:00.000-07:00

The Developing Expert Voices was an okay experience. It made my understanding better in different areas such as transformations because before I would get things mixed up all the time. The one thing I know I will not forget is stretches before translations. Doing the trigonometric identities was hard for me because I missed the very first day, which started everything. It was difficult to grasp but doing the project I could do better and find patterns much faster. The logarithms were difficult to explain because you somewhat just have to know what values are what and which formula to use. Overall, it was okay. I think I would have had a better experience if I did not waste so much time. I also had difficulty accessing a computer because of my stubborn one. I need a video card because mine is lacking memory in graphics, therefore monitor did not show any picture and said “no signal” and turned off itself. That was difficult to deal with, but hey, I finished! That’s the best part!

Question #1

2007-04-25T18:13:00.000-07:00

George is a boy. On Friday George went over to a friends house to play five card poker. There were six of them that played, including himself. After a long two hours the winner was narrowed down to him and a friend. He was in the last two. George could tell that this game was dragging on way too long by the way everyone looked. The other four sat there bored out of there minds, so he decided to go all in. Fortunatly, him and his friend were even in chips. George's friend showed his hand first and had a full house of 7's over queens, suited. George went in blind and did not look at his hand. Quickly, he grabbed a pen and paper and calculated his chances of beating a full house of 7's over queens that is suited.

Solution #1

2007-04-25T18:07:00.000-07:00

Okay since George's friend has two queens there are only two chances of him getting a royal flush because the two queen's are already used and a royal flush consists of five consecutive cards from nine to king in the same suit. So George wrote out:

ROYAL FLUSH:
(2C1)(5C5)

(2C1) because there are only two suits left to make a royal flush since the two queens are used up.
(5C5) because there are five cards to choose from and the five cards have to be consecutive from nine to king.

STRAIGHT FLUSH:
(2C1)(6C1)(4C4)

(2C1) because you choose a suit.
(6C1) because there's six cards that can create a straight flush.
(4C4) because you could choose four of the cards around the card you chose from 6C1.

you choose a suit, then you choose one of the numbers from A to 6. Then you choose the 4 numbers around it. Example: for the suits, clubs and diamonds, both 7 AND queen are missing. So, the only straight flushes that can be made are the ones from A-6. For the 3rd suit, Clubs, ONLY the 7 is missing. So, you don't even have to choose a suit, for it's already been decided. So then, you have A-6 and 8-K. For the 4th suit, NO CARDS are missing, so you can pick one of 13 numbers and pick 4 of the 4.

FOUR OF A KIND:
(11C1)(4C4)(10C1)(4C1)

(11C1) because since the queens and sevens are taken out then that means that there are only 11 other possible face cards that can be four of a kind.
(4C4) you need all four cards of the same face card.
(10C1) because there are 10 cards left for the kicker to be, so choose a face value for the kicker.
(4C1) the suit that the kicker will be.

HIGHER FULL HOUSE:
(6C1)(4C3)(10C1)(4C2) + (6C1)(4C3)(1C1)(2C2)

(6C1) because there are only 6 other cards that can beat the seven's. So, choose 1 of the 6 face cards that are higher than the seven.
(4C3) the face card determines which other cards you need to get. So, you need three of the four face card.
(10C1) there are ten available cards to be the pair because the sevens are used up and queens and what you have that made the three of a kind.
(4C2) because of the four cards of that face you only need two of them to make it a pair.

(6C1)(4C3)(1C1)(2C2)
choose a number for 3 of a kind, then get 3 of those numbers, then choose a pair, which have to be queens, then make it's pair

CHANCES OF BEATING:
((2C1)(5C5) + (2C1)(6C1)(4C4) + (11C1)(4C4)(10C1)(4C1) + (6C1)(4C3)(10C1)(4C2) + (6C1)(4C3)(1C1)(2C2))/ (47C5)

(47C5) is divided because it is all the possible hands that he could make with the remaining cards.

ANSWER:
1928/ 1533939 = 0.0012569

Question #2

2007-04-24T21:01:00.000-07:00

It was a hot and sunny Sunday. George was at home bored. He noticed that his hamster, Joung, was running on his excercise wheel. George was curious to know when the wheels is at its maximum and minimum height. So, he grabbed a red string and tied it to a point. He measured the diametre and found out that it was 6 inches in length. He timed Joung and recorded that Joung did 122 revolutions every minute.

A) Sketch a graph of the height of the string as a function of time, starting at t= 0 seconds.
B) Write a sine and cosine equation for the function.
C) How many radians did the wheel turn in 30 seconds?

Solution #2

2007-04-24T20:40:00.000-07:00

SOLUTION

A) To sketch the graph George has to find out what the period is. The information that was given was that the wheel did 122 revolutions in 1 minute. What George does with the numbers is he divides 1 by 122 because that’s how he’ll find out the period. With that number, 0.0082, George will have to multiply that by 60 to get how long each revolution takes. What George just found out is B. He remembers that the period is equal to 2π/ B. So, all George has to do is substitute 0.492 for the value or B and he has the period, which is 2π/ 0.492. He remembered that there’s usually four different numbers on the x-axis. What he does to get the other values is he divides 0.492 by two and gets 0.246 that will by half of the period. He divides 0.246 by 2 and gets 0.123, that will be ¼ the period. He adds 0.123 and 0.246 to get ¾ of the period.

B)To generate an equation for Cosine George will have to look at what the basic cosine graph looks like. Usually, George starts by drawing the cosine graph in dashes and not solid lines because that’s not the graph he’s looking for it’s just there to help him figure out how to generate and draw the graph. He notices that the cosine graph starts at its maximum height, since he wanted to find out what it’s at t=0 the graph would have to start at a minimum. That would mean that cosine would have to be negative or the value of A would be negative so that the cosine graph flips vertically and it starts at a min. After that, George remembers in class that stretches are before translations. He notices that the maximum height of the exercise wheel is 6 inches and the minimum is 0 so he adds those numbers up and divides by 2 to get the average. Now he knows what the amplitude is, it is three. George already figured out what the period was, which is 2π/ 0.492. All George has to do now is shift the graph up 3 units so that it starts at t=0. Moreover, there he goes, he figured out the cosine equation.

REMEMBER: f(x)= AcosB(x-C)+D

A= -3
B= 2π/ 0.492
C= 0
D= +3

To generate the equation for Sine George draws out the basic Sine graph just as a reference. It already starts at a minimum but he’ll have to stretch the graph vertically by three units because that’s where the average is. The period is still 2π/ 0.492. Now George has to shift the graph 3 units up.
Since the graph doesn’t start at 1=0, he’ll have to shift the graph. The closest point is on the left so he’ll have to shift the graph to the right. George remembers to watch the signs of C because when its negative it shifts to the right and when its positive it shifts to the left, but George wants the graph to shift right so C has to be negative. C is equal to -3π/ 2. The last thing to do is to shift the graph up three units and he is done.

REMEMBER: AsinB(x-C)+D

A= 3
B= 2π/ 0.492
C= -3π/ 2
D= + 3

C)To find how many radians the wheel turned in 30 seconds George has to divide 122 by two because that’s how many revolutions it did in 60 seconds. Therefore, to find how many revolutions it turned in 30 seconds we have to divide 122 by 2, which gives him 61. To find out how many revolutions it did in radians in 30 seconds he’ll have to multiply 61 by 2π because 2π is a full revolution. That turns out to be 122π.

Question #3

2007-04-22T20:29:00.000-07:00

George got sick. His doctor gave him all the information. On the first day of him being ill the initial count of the bacteria was 5000. His initial white blood cell count was 2000. On the 8Th day the bacteria count was 4522 and his white blood cell count was 3769. Find the model of the white blood cells and bacteria.

Solution #3

2007-04-22T20:09:00.000-07:00

George used this formula because he has not given the period or the rate or number. Therefore, by using this formula all he needs to find out is the just the model.

A= the final value you end up with
A = the initial value you start with
T= 8-1, because that was the time of the initial and final count.
Model= is the unknown or what George is trying to find out.

The formula is used to find out both the bacteria and white blood cell model. George could check if hes answer is correct by substituting the model in. The answer on the right should be very close to the answer on the left side.

Question #4

2007-04-22T15:54:00.000-07:00

It was getting late and George knew that he still had math homework to do. He finished most of it with no problem. There was this one question where he wanted to raise his hands in defeat. But, he noticed that it was just trigonometric identities. He took out his math dictionary and looked at the identities.

Solution #4

2007-04-22T15:51:00.000-07:00

SOLUTION

STEP 1: Pick one equation to work on first.
STEP 2: Write equation out on a different paper.
STEP 3: The equation in the numerator is CSC^4x - COT^4x. George notices that by expanding the equation, he'll be able to reduce with the equation in the denominator because they make 1. He'll expand it to (CSC²x-COT²x)(CSC²x+COT²x).
STEP 4: Reduce (CSC²x+COT²x) from the numerator and denominator.
STEP 5: By reducing (CSC²x+COT²x) from the numerator and denominator what George will be left with is CSC²x-COT²x +COT²x.
STEP 6: George notices that -COT²x +COT²x make a zero pair. Therefore, CSC²x is left.

STEP7: Now that George finished solving for one given side he could start on the other.
STEP 8: Write equation out on a different paper.
STEP 9: George notices that the equation in the denominator is also a Pythagorean identity where 1 + TAN² x = SEC²x, but in George's case it is rearranged into where TAN² x = SEC²x -1.
STEP 10: George also notices that SEC²x is a reciprocal identity where SECx = 1/ COS²x. He also notices that TAN²x is equal to SIN²x/COS²x. Now what George has is 1/ COS²x/ SIN²x/COS²x.
STEP 11: When a fraction is divided by another fraction what you do is you keep one of the fractions the same and multiply that by the reciprocal of the other fraction. So now what George has is ( 1/ COS²x)(COS²x/SIN²x).
STEP 12: When you multiply fractions you could reduce them diagonally because when you multiply it it's going to reduce anyway because it becomes 1 and when you multiply anything by one it's whatever you multiplied it to be.
STEP 13: Now what George is left with is 1/ SIN²x which is equal to CSC²x, it's a reciprocal identity.

STEP 14: Now that George has found the missing given sides he could now start working on finding what Z is equal to.
STEP 15: Since he knows the missing sides and the triangle is a right triangle, he could use the pythagorean theorem to solve for Z. The pythagorean theorem is a² + b² = c². So, now George could substitute the values CSC²x into a and b.
STEP 16: What George has now is (CSC²x)² + (CSC²x)² = Z².
STEP 17: CSC^4x + CSC^4x = Z². Once George adds them he gets 2CSC^4 = Z².
STEP 18: To get rid of the square on Z we'd have to square both sides.
STEP 19: So Z is equal to √2CSC^4