Solution - Episode 1: Conic Section

A conic section is defined by the equation 25x² + 4y² - 100x + 8y + 4 = 0

Answer the following given the equation above

a) What is the conic section?
B) write the equation in standard form
c) sketch the graph

Now to solve:

To solve these question first thing that i would do is question "B", because you need the standard form of the equation to know what it is and also sketch the graph.

B) Write the equation in standard form.

The First thing you should do is collect like terms so it should look like this.
25x² - 100x + 4y² + 8y = -4

The next would be to factor out 25 out of the x values it should look like this.
25(x² - 4x) + 4y² + 8y = - 4

The next would be to factor out 4 out of the y values it should look like this.
25(x² - 4x) + 4(y² + 2y) = - 4

Now we have to complete the squares. (AND DON'T FORGET IF YOU ADD ON ONE SIDE YOU HAVE TO ADD IT ON THE OTHER SIDE TOO AND ALSO DON'T FORGET TO MULTIPLY OUT WHAT YOU ADD. Ex. 25(x² - 4x + 4) the four is actually 100.)

25(x² - 4x + 4) + 4(y² + 2y + 1) = - 4 + 4 + 100

After you have done that you have to multiply all the terms by the reciprocals of the coefficients so it should look like this.



And there you go now have put that equation into standard form.


Now for Question A)

A) What is the conic section?

Well the answer is that it is an ellipse. You can notice that it is an ellipse because the denominators are different.

NOW to graph C)

C) Sketch the graph

TO sketch the graph you first find the center which in this case it is (2, -1)
Next you would check with denominator is larger so you can determine if it is a vertical or horizontal ellipse. Well this ellipse is a vertical ellipse.
after you find the semi-Transverse axis and the semi-minor axis. to find that you need to take the square roots of each denominator. Which are 2 and 5.
After you have that you start from the center and go 2 units from to the left and 2 unit to the right and put points at those spots. That should give you the minor axis.
After you have that you start from the center and go 5 units up and 5 unit down and put points at those spots . That should give you your transverse axis.
Now that you have those points just connect the dots and Ta-da you have sketched the ellipse

Should look something like this

Solution - Episode 1; Combo Question

The question is find the third term of the value (b ² - (2/b))⁷

The first step is to let b ² = a and let -2/b = b

The second step is in put values so you get ₇C₃ a⁴b ³

The Third step would be to substitute the original values back in to the question.

The fourth and final step would be to solve the rest algebraically.

If you want to see a visual of how it is suppose to work out there is an image below for the answer.

Solution - Episode 2; The Warthog and Mongeese

The question:

There are seven students going on a journey together and they need to get seating arrangements completed To ride in they have one warthog ( which can seat 3 people) and two Mongeese (which can hold 2 people each). If Richard, Lawrence and Justus all have to sit together, and Roxanne can sit with anyone, how many ways can everyone be seated?

Remember, where each person sits DOES matter, and the mongeese themselves are distinguishable.

The Solution.




The answers are above this very text and if you double click it it will enlarge i think. but if it does not work i will guide you on how to do it. (HINT: I would recommend to click the image cause there are some diagrams)
Step 1.) Find the number of ways to seat Lawrence, Richard , and Justus
- All must sit together so therefore they must be in a warthog
- Since order does matter, the number of ways to seat them is "3!" which is equal to 6

Step 2.) Find the number of ways to seat Roxanne and the other students.
-The two seat ones (s1), and the two seat two's (s2) are non-distinguishable objects therefore the number of ways to seat the students in the mongeese is 4 factorial (4!) all over 2 factorial (2!) times 2 factorial (2!) which will give you 6

Now for step 3

Step 3.) Total number of ways to seat everyone is (# of ways to seat Lawrence, Richard, and Justus) ( # of ways to seat Roxanne and the rest of the students.

(6)(6) = Which the total ways to seat everyone is 36. Ta-Da!

Solution - Episode 3; The Identity

Hey guys, it's Lawrence here, and I'm gonna give you the solution to the second question in Episode two that had to do with the Identities.

So, because we're solving for identities, we draw the "Great Wall of China" perpendicular to the equal sign and NEVER cross over it.

So to solve it, we'll work with the left side because it'll be easier to "decode".

So FIRST, we'll expand Cos(x+y) * Cos(x-y) so we can see a better picture. Using the infamous "Sine Dance" Cos(x+y) becomes (cosXcosY - sinXsinY) and expand Cos(x-y) to become (cosXcosY + sinXsinY).

When we complete that, the identity now looks like;
(cosXcosY - sinXsinY) * (cosXcosY + sinXsinY) = Cos²X - Sin²Y

From there, we'll proceed algebraically and do the multiplication on the left side of the "Great Wall". We will multiply so that we get:
cos²Xcos²Y + cosXcosYsinXsinY - cosXcosYsinXsinY - sin²Xsin²Y

If you haven't already noticed, there is some canceling going on here.

cos²Xcos²Y + cosXcosYsinXsinY - cosXcosYsinXsinY - sin²Xsin²Y

and we then get...

cos²Xcos²Y - sin²Xsin²Y = Cos²X - Sin²Y

So, from here, we observe the equation and look at what we need to get to solve the identity. Look at the right side and you'll notice it's Cos²X - Sin²Y. On the left side, we already have the Cos²X and the Sin²Y, it's just "with" another. So because we want those ones, we'll use the trig identities to switch cos²Y into (1 - Sin²Y) and sin²X into (1 - Cos²X). In which we then get:
cos²X(1 - Sin²Y) - (1 - Cos²X)sin²Y = Cos²X - Sin²Y

And we do the multiplication on the left side again.

cos²X - cos²XSin²Y - sin²Y - Cos²Xsin²Y = Cos2X - Sin2Y

We now have more cancelling we can do.

cos²X - cos²XSin²Y - sin²Y - Cos²Xsin²Y = Cos2X - Sin2Y

And the identity is now complete:

Cos²X - Sin²Y = Cos2X - Sin2Y






Remember your trig identities as it can help to solve identities like these. Also remember there are more than one way to solve identities so try to experiment sometimes.

Rence ~ Out

Solution - Episode 3 ; 2 teams of 7

Given that there are 4 red, 6 blue, and 3 green players to choose from What is the probability that you'll pick at least 2 blue players for your team, when teams are chosen random.

Treat player groups (colors) as non- distinguishable

to Solve the first thing you should do is find the sample space which in this case it is


now that you have the sample space you take the probability of the number of blue players is greater or equal to 2 and you will get the correct answer.

Credits

Mathematics is the Science of Patterns

Team LRJ Studios

Production Unit

Director............................................................. Justus

Story ..................................... Lawrence, Richard, Justus

Planning.................................. Lawrence, Richard, Justus

Producer........................................................... Richard

Executive Producers ............................. Lawrence, Justus

Music Director .................................................... Justus

Editing ............................................................. Justus

Game Play ............................... Lawrence, Richard, Justus

Computer Tech ..................................................... Eric

Advertising ................................................... Lawrence

Cast

Lawrence ...................................................... Himself

Richard .......................................................... Himself

Justus ........................................................... Himself

Mr. Kuropatwa ................................................. Eric T

Guardian # 1 ................................................... Justus

Guardian #2 ................................................ Lawrence

Guardian #3 ................................................... Richard

Guardian #4 ..................................................... Eric T

Guardian #5 .................................................... Justus

Guardian #6 ..................................................... Eric T

Programs / Hardware / Software Used

Sony Vegas

Audicity

Adobe After Effects

Xbox 360

Halo 3

Music

Boys Like Girls - Five Minutes To Midnight

Daft Punk - Better, Faster, Stronger

Daft Punk - One More Time

Lupe Fiasco - Superstar

Backstreet Boys - I Want It That Way

Colbie Cailait - Bubbly

Teriyaki Boys - Tokyo Drift

DJ Tiesto - Love Comes Again

Halo 2 & 3 Soundtrack

Time Management

Hours Spent ; 50 +

Sleep ; Close to none

Story ; 3 Hours

Game Play ; 10 + Hours

Editing ; 20 + Hours

Xbox 360 Red-Rings of Death ; Occurred Twice

Special Thanks

Eric T
Roxanne Y

Francis B
Mr. Kuropatwa

... And YOU, for viewing our project :)

And I wonder, if you know... What it means... To find your dreams come true