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 "Heat Wave" in Winnipeg

Hey folks, it's Lawrence here again, and I'll be solving the Transformation question.

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

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.