Monday, June 2, 2008
MITSOP Episode 1
Solution - Episode 1: Conic Section
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.
25x^{2} - 100x + 4y^{2} + 8y = -4
The next would be to factor out 25 out of the x values it should look like this.
25(x^{2} - 4x) + 4y^{2} + 8y = - 4
The next would be to factor out 4 out of the y values it should look like this.
25(x^{2} - 4x) + 4(y^{2} + 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(x^{2} - 4x + 4) the four is actually 100.)
25(x^{2} - 4x + 4) + 4(y^{2} + 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 first step is to let b ^{2} = a and let -2/b = b
The second step is in put values so you get _{7}C_{3} a^{4}b ^{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 "Heat Wave" in Winnipeg
During episode two, the Guardian asks these Questions:
An Average daily maximum temperature in Winnipeg follows a sinusoidal pattern with a lowest value of 8.4 on April 7th, and a highest value of 30.9 on August 4
A) Sketch the graph and describe the variation with 2 equations
- a sine and a cosine function.
B) What is the expected Maximum temperature for July 13th?
C) Algebraically determine how many days will have an expected maximum of 23.0 or higher
A) Sketch the graph and describe the variation with 2 equations
- a sine and a cosine function
So we'll start with the graph since we need to find the equation to solve B and C anyways. This function is described by the equation t(d) = A (sin or cos) [B (d - C)] + D ; Where t is Temperature, d is Days (time), A is the Amplitude, B is the Period, C is the Phase Shift and D is the Sinusoidal Axis.
So to start off, we'll find D using the maximum and minimum value to find the Sinusoidal Axis. Add 30.9 and 8.4, and then divide it by two to get the middle.
D = 30.9 + 8.4 / 2 = 19.65.
Great, now we have our D value! From here, we can find the A value. We do so by taking the maximum value and subtracting the sinusoidal axis so that we find the amplitude of the wave.
A = 30.9 - 19.65 = 11.25
Ok! So we now have the A and D values! Next, we'll find the B value, which is the period. B is defined by 2π/period, so we'll find that by having to add up all the days.
First, we'll have to start at April the 7th, because that's when our lowest value occurs earliest in the year. So we find the sum of the days up until August the 4th. So we go;
Apr .7 + May 31 + Jun. 30 + Jul. 31 + Aug. 4 = 103
Because that's only a fraction of the period, we have to find the next 3 points where the temperature hits the sinusoidal axis, the maximum and back to the sinusoidal axis. So from 103, we have to add the month's we haven't yet. It's not like they didn't exist so we have to count them in the period. So we'll have to add the 23 remaining days in April, the 31 days in March, the 28 days in February, and the 31 days in January. So...
103 + 27 + 31 + 28 + 31 = 216
That's only half of the period, so we'll have to multiply it by two.
216 * 2 = 432.
We now have the B value which is 2π/432. So far we have
A = 11.25 B = 2π/432 D = 19.65
Next, we'll have to find C if we want to fully graph this thing. So because our lowest value occurs on April the 7th, we can find the Phase Shift for the Cosine Function of this Equation. So basically, we have to add up the days from the first day in the year, until April the 7th. Therefore...
31 + 28 + 31 + 7 = 97.
For the cosine function, the Phase Shift is 97, but to find it for the Sine Function, we'll have to add it to the first quarter of the Event. To do that, we'll take the Period, which is 432, divide it by two to get half, which is 216, and the divide it by two again to get a quarter, which is 108. We will then add the 97 to 108 to find the Phase Shift of the Sine Function.
108 + 97 = 205.
We must also add 97 to the half point and the third quarter of the event if we want to properly draw the graph out since it THERE IS A PHASE SHIFT.
216 + 97 = 313
324 + 97 = 421
GREAT! Now that we have all our "tools" we can now write out the sine and cosine function for this event, and draw it as well.
A = 11.25, B = 2π/432, C = 97 or 205, D = 19.65
SINE FUNCTION: t(d) = 11.25sin[2π/432(d - 205)] + 19.65
COSINE FUNCTION: t(d) = 11.25cos[2π/432(d - 97)] + 19.65
And this is what the graph would look like (Click the image)
B) What is the expected Maximum temperature for July 13th?
To solve this one is pretty straight forward. You want to find how many days you have for 'd' to plug into your function, and then just go from there. So we have to add all the days up until July the 13th.31+28+31+30+31+30+13 = 194
So now we plug it into the Function and also, we'll use the sine function because we won't have to deal with negative values. t(194) = 11.25sin[2π/432(194 - 205)] + 19.65.
There's nothing special really here. You just plug it into your handy dandy calculator and you come up to an answer of 19.6185, which we can round to 19.62 degrees Celsius.
C) Algebraically determine how many days will have an expected maximum of 23.0 or higher.
Now, when we solve this one, there's a little trick in how you treat the function so that you don't end up messing yourself up. Okay, so we want to find how many days will have an expected maximum of 23 degrees Celsius or higher.
First, we'll let θ = [2π/432(d - 205)], just so that it's easier to deal with. So the function ends up looking a little like this: 23.0 = 11.25sinθ + 19.65. From there we'll subtract 19.65 from both sides. We then get 3.35 = 11.25sinθ and we'll reduce 11.25 by dividing it from both sides. We then get 0.2977 = sinθ. We'll use the Arc sine function on our calculators to isolate θ.
After that, we get 0.3022 = θ. Place back in [2π/432(d - 205)] and we'll then have 0.3022 = [2π/432(d - 205)].
Multiply the reciprocal of 2π/432 from both sides and we'll have 20.7890 = d - 205. Add 205 to both sides to isolate d and find our answer in Days.
get a final answer of 225.7890. We can round that to 226 days or even change the decimals into hours and minutes.
And that's it folks!
Rence ~ Out
Solution - Episode 2; The Warthog and Mongeese
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
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) = Cos^{2}X - Sin^{2}Y
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:
cos^{2}Xcos^{2}Y + cosXcosYsinXsinY - cosXcosYsinXsinY - sin^{2}Xsin^{2}Y
If you haven't already noticed, there is some canceling going on here.
cos^{2}Xcos^{2}Y +
and we then get...
cos^{2}Xcos^{2}Y - sin^{2}Xsin^{2}Y = Cos^{2}X - Sin^{2}Y
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 Cos^{2}X - Sin^{2}Y. On the left side, we already have the Cos^{2}X and the Sin^{2}Y, it's just "with" another. So because we want those ones, we'll use the trig identities to switch cos^{2}Y into (1 - Sin^{2}Y) and sin^{2}X into (1 - Cos^{2}X). In which we then get:
cos^{2}X(1 - Sin^{2}Y) - (1 - Cos^{2}X)sin^{2}Y = Cos^{2}X - Sin^{2}Y
And we do the multiplication on the left side again.
cos^{2}X - cos^{2}XSin^{2}Y - sin^{2}Y - Cos^{2}Xsin^{2}Y = Cos2X - Sin2Y
We now have more cancelling we can do.
cos^{2}X -
And the identity is now complete:
Cos^{2}X - Sin^{2}Y = 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
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.
Lawrence ; The Reflection
To start off, my rundown. At first, this whole deal was gonna be a "class room" setting kinda deal. Or something a long those lines. But due to our very busy lives, we had to simplify our story and cut it into 3 episodes instead of the former 6 for each question that we basically had. Because of our limited time, and conflicting schedules, we didn't get to set out to what we basically were going to do, from our experience just filming the intro. YEAH, the introduction was that DIFFICULT to film. So we had to break 'em down. Once we started getting the flow of things, the next episodes started to fall into place a lot easier, because well, basically we knew what we were doing. Once the filming was done, we just got started. Intertwining the questions was going to be easy enough work. The math itself? Not so much.
With the math, we basically had simple questions intertwined with each other. So really, some questions had pre-questions you had to solve, before actually solving the question. In a sense, we did not make it easy on ourselves, which has it's pro's and cons, but hey, things worked out pretty fine. Just to make sure our math was correct, we ran it by Mr. Kuropatwa once, and with the help of our friend, Roxanne, she double checked it for us, and for that, we are grateful.
Inputting audio was most likely, one of the best parts of doing this project, probably because we had WAY too much fun with it. We ended up saying stuff that was WAY too funny, and WAY to ridiculous to put into the film to ever take it seriously. So, you can basically expect A LOT of bloopers and outtakes when our whole project is finish. Due to us being a three man crew, and having more that one character and since we're not professional voice actors and can make our voice into like, 6 separate characters, we were graced with the help of our good, and long time friend Eric. It was hard enough to record audio without Eric, but he just made it even harder, because he would say the most ridiculous things, and we'd have to re-record the scene ALL OVER AGAIN. But I mean really, let's get to the real reflection.
- Why did you choose the concepts you did to create your problem set?
Why is a good question to start off with. I chose, for my questions anyway, the concepts of sine functions and identities mostly because I had the most trouble with them. I mean like, for the first test I scored a 34% basically because I didn't understand how to use solve the sine function. Identities, I faired more easily with, but in my mind, it was still like scrambled eggs, so I still had to wrap my mind around that properly. Even though it was one of my weaknesses, I feel that I have created it to become one of my strengths with this project. With the whole identities thing, it's basically the way it is, they all have different identities so, it's hard to understand the massive amount the way it could be solved. The way that Mr.K thinks makes me wonder how you can solve things more efficiently, and in different ways, even if they're created within your own mind, because he showed me that, you don't always have to do it the generic way that teachers just plop an equation on you and say "Go solve this." That unit showed me that there are multiple ways to solve equations. So basically I'm happy that I went with the concepts that I did, because now that I completely understand them, I'll be more confident in the future solving these kind of problems. It was also a great learning experience while I was creating the question, because of the different ways that you can look at it.
- How do these problems provide an overview of your best mathematical understanding of what you have learned so far?
I always look at things in the way that, if it's the hardest thing for you to do, once you master it, it becomes one of your strengths. So because we did questions that we all had problems with understanding, it eventually made us have to pull out our best in thinking and show that we understand the question and use our best math to solve it efficiently even though we were still learning to understand it in the process. There's never a time where, I can look at something and say that's my best work, because if I'm satisfied with it, then I know I didn't do my best, because I'll always be looking for a better way to do it and solve it. So because a lot of these problems we chose were pretty hard, we had to use our knowledge of our mathematical understanding to figure it out, and learn during the solving phase. Also the fact that exams are coming up, helps me touch on things that we've done early on in the year, and refreshes my mind to be ready for that big provincial exam, in the event (like most people I presume) that we forget some of the things that we learned, since we're not constantly doing what we first learned in the semester.
- Did you learn anything from this assignment? Was it educationally valuable to you? (Be honest with this, if you got nothing out of this assignment, then say that, but be specific about what you didn't like and offer a suggestion to improve it in the future.)
Okay, let's be honest now. I don't know if this can be properly stated as an "Assignment", especially when it's worth about 20% of our grades. I will definitely say that I've learned things from this assignment, mostly because I've had to dig back into February's notes, and experiment in the solving process of the questions that we had created. After some of the suggestions Mr. Kuropatwa made to modify the questions, we started to look at the questions in a different light. It was definitely an education eye-opener, and made me see math in a way that I've never seen it before. Also the fact that, Mr.K's been the FIRST math teacher I've ever had fun doing math with [subtract the stressing tests and the ones I failed ): ], I actually felt comfortable learning the math, especially with his innovative ways of teaching. But back on topic. Doing the project definitely served as a review and and educational help in solving these units.
I'll admit, when we started the project, I had so many ideas circulating in my mind, that I just couldn't seem to get it on paper. After seeing Justus' Halo Montage, I decided that we could do something along those lines. At first, we were gonna use the math we learned this semester to solve the ways the scientist's enhanced the "Spartans" in Halo. It seemed like a really good idea, as it involved a lot of math, but we decided to go along with a more student relating journey, and so we created a class room storyline, but we ended up not having enough time to complete such a task, so another cut was made and we had a journey of three students, under their Master JabbaMathee, go under instruction to solve math questions. It was long and difficult, but we got it done, and in amazing time at that (We did about 80% of the work in 3-4 days out of the many days we worked on it ) Not only was this a good group experience, it was also a great learning experience. This project has helped review and go over the math I failed to understand, because after all, Math is the Science of Patterns.
Reflection: Richard
Why did you choose the concepts you did to create your problem set?
The reason we decided on these particular questions was because we (mostly me) had problems on them. The question that a helped come up with were the Choosing teams, while blindfolded question and the ellipse question. The reason i picked that particular choosing question is because it was one of the most confusion concepts of the whole year. This sub unit was probably the hardest because I missed a class for an English Field trip.
This problems provide an overview of our best mathematical understanding of what we have learned so far because our questions were from various units, not just two or three. This project also created our weaknesses into strengths (well for some of us).
- Did you learn anything from this assignment? Was it educationally valuable to you? (Be honest with this. If you got nothing out of this assignment then say that, but be specific about what you didn't like and offer a suggestion to improve it in the future.)
Justus' Reflections for DEV! wewt
- Why did you choose the concepts you did to create your problem set?
We all choose our problems based on two parameters, a.) what we had previously had the most trouble with, and b.) What would fit best into our videos story line. So initially we tried to make the questions based on the storyline, and when that didn't work we said, "hey, lets just make questions that are hard. So we did that. After we made the questions we worked the storyline around them in the perfect way; such that the questions had no effect on the story at all! (Except that they were an object in the story.) So yeah, in all honesty, I just choose my questions based off units I thought were interesting, or I was having difficulty with, such as the Warthog question.
- How do these problems provide an overview of your best mathematical understanding of what you have learned so far?
Well they definitely opened my eyes to what I need to practice more for the upcoming exam. As was said during an earlier class, creation is the highest level of learning. I found this to be entirely true, as when making the questions, having them solve neat and nicely was immensely difficult. You had to look at exactly which parts of which questions changed what in the outcomes, so that it was actually solvable. In the end I was able to see the question (types) in a whole new light.
- Did you learn anything from this assignment? Was it educationally valuable to you? (Be honest with this. If you got nothing out of this assignment then say that, but be specific about what you didn't like and offer a suggestion to improve it in the future.)
In terms of mathematics I learned quite a bit also. I deepened my understanding of conic sections, probabilities, trig, and loooootttsss about solving questions by looking at them from different perspectives (block of wood anyone?).
It helped deepen my understanding of mathematics profoundly because to create and solve the questions on here, I had to find and exploit the patterns in the questions themselves.
Overall I'm quite glad I did the project, and although I feel I could have done better, am happy with the overall outcome. :)
Alrighty, Justus, the purple spartan, out
ciao!
Credits
Eric T
Roxanne Y
Mr. Kuropatwa