Application of polar coordinates. Batteship

In Mathland, we used our knowledge of polar coordinates to play a game if battleship. It was actually fun

Rules to battle ship

1. Set up your four ships on the coordinate graph. The points have to be adjacent to each other not diagonal. Don't show the opponent 

2. Each player gets one guess per turn. He declares a specific point on a graph (r,theta) and the other player will tell whether you hit or not. He must also tell you if the ship sank. A shirt sinks when it has been hit according to the length of it.

Highlighted would mean my ships or hits

Graphs of polar equations

Graphing polar equations is much different than graphing regular (x,y) equations. 

Let's say we were given the equation: r=4sin theta

Now remember that for sin, the value of theta is between 0<theta<2pi

By substituting various values of pi into the equation, we can also figure out the value of r for each calculated value of theta. Remember to plot the points according to polar coordinate system. So if theta is pi/6, r would be 2. That particular point is (2, pi/6)

Symmetry

1. Line theta = pi/2: replace (r, theta) with (r, pi - theta) 

2. The polar axis:    Replace (r,theta ) with (r, -theta)

3. The pole:            Replace (r, theta) with (r, pi + theta)

Let's try to see if the equation: r=3+2cos theta has any symmetry

2. R = 3 + 2cos(-theta)

    R = 3 + 2cos theta. 

So it is symmetrical to the polar axis. 

To find the maximums of |r| we have to plug in different values of theta according to the range and see which value of r is the highest.

Shapes: 

10.6 polar coordinates

In addition to graphing points in terms of (x,y), we can also do so through (r,theta) . When we do this, his is called the polar coordinate system. R is the directed distance while theta is the directed angle. To express r, we generally draw circular rings around the origin. If the point is located on a further ring, we have a higher value if |r|. For instance of r =3, then the point will be located on the third ring. The exact location on the ring is determined by the angle. Theta is expressed through radians. We can use the angles on the unit circle to express the angles in the polar coordinate system. 

Now. How do we even convert (x,y) to (r, theta) and vice Versa?

To convert to (x,y)

X = r cos theta

Y= r sin theta

 To convert to (r,theta)

Tan theta = y/x

r^2=x^2+y^2

Now let's try to convert (3,pi)

X=3 cos pi = 3

Y = 3 sin pi = 0

(3,0) 

Congrats this is the main gist of section 6

Rotation and systems of quadratic equations

Okay... Let me be frank. This section is going to make you cry, because it definitely made me. BUT! With my guidance you are going to understand this so easy! 

Let's start with the equation of an xy-plane.

Ax^2+Bx^2+Cy^2+Dx+Ey+F=0

The objective of rotation is to eliminate the xy term because it completely messes up our graphing process! We do so in e following steps.

1. Find the angle in which we rotate the graph through the equation..

    Cot 2 theta = A - C

                            B

2. Use the angle to find x and y, in the equations

    X= x' cos theta - y' sin theta

    Y= x'sin theta + y' cos theta

3. Substitute x and y into the original equation. ( we will also try to rearrange this into the equation of a parabola, ellipse, or hyperbola). 

Remember that the general second degree equation is 

A'(x')^2 + C'(y')^2 + D'x' + E'y' + F' = 0

4. We graph this new second degree equation. The angle of rotation is the angle we found earlier.

Here's an example of how to do a rotation

Hyperbolas

 Hyperbola is defined as a set of all points (x,y) the difference of whose distances from two distinct fixed points (foci) is constant. 

The equation is given by so (standard form)

Going horizontal: (x - h)^2  - (y - k)^2

                               a^2.           b^2 

Going vertical:     (y - k)^2  -. (x - h)^2

                               a^2.            b^2 

Foci: c^2 = a^2 + b^2

Notice how, unlike ellipses, a doesn't denote the major axis. This is due to the fact that such things are not used here, a and b stay in the same place regardless of anything. The direction in which a hyperbola goes (horizontal or vertical) depends on whether x or y is the first term. If y is the first term, the hyperbolas go along the y axis. Opposite is true if x is the leading term, 

Center (h,k) 

Vertices:

 if hyperbola is going horizontal (x leading term): (h+a,k) (h-a,k)

If hyperbola is going vertical (y is the leading term): (h, k+a)(h, k-a) 

Remember for the box that we add b to the h or k to find the boundaries.

As for the foci

If hyperbola going horizontal: (h + c, k) (h - c, k) 

If hyperbola going vertical: (h, k + c) (h, k - c)

Asymptotes

Horizontal: y=k +/- b/a (x - h)

Vertical: y = k +/- a/b (x - h)

Here's an example of his to make a hyperbola.

My application of math in a spiritual way

What's up mathland!? I serriously could not find any topic to do a math post on, but I suddenly remembered that we did an extra credit assignment last semester, that did apply math to Christianity.
Now of course, remember than no virtue or element of Christianity can be calculated in terms of math, because they are all aspects of the heart, not the mind.

Even though we cannot describe God or the universe with mathematically calculate functions or equations, we can truly create analogies that may be of more value. These analogies could possible be how our life works. The function y=mx+b describes the equation of a line. Let y be our continuous line. It is important to know that even though our life ends at a certain point, it will always continue in the after-life, just like how a line has no beginning and no end. Now, let mx+b represent the factors that affect y. Life has its  “ups” and “downs” similarly how a slope can be positive or negative in our basic equation. Now, let x be the decisions we make in our life. The degree of a slope depends on the value of x. Similarly, the degree of our “ups” and “downs” in our life is totally dependent on the decisions we make in our life. Therefore, our general life is completely dependent on our actions. However, allow b represent a divine intervention or influence. In a line, the value of b either lowers or raises the y-intercept. Similarly, if we allow divine intervention enter our lives, our lives can be much better or positive on the graph. However, even with divine intervention, our decisions may be negative and our lives will become even more terrible despite intervention. Therefore, humans must make positive decisions in order to live out the best of their capabilities.
So how about this philosophy!? Please comment on this opinion. It's a little bit out of the blue.

Ellipses

An ellipse is defined as a set of all points, the sum of whose distances from two foci is constant. It is also important to identify the different parts of a parabola. First of all there is a major axis that can go either horizontally or vertically. The smaller axis is the minor axis. 

The equation of an ellipse is given by:

The major axis is "a" and is always bigger than b. If it's under x, then x is the major axis. The same is true for y. 

C^2 = a^2+ b^2

The center is (h,k) 

We can find the vertices by knowing the value of a. Once we know it, we add the value of a to the y or x points of the center. 

For the foci: we add and subtract the value of c to the coordinates of the center.

Here's an example of an ellipse

Parabolas

A parabola is defined as a set of all points that are equidistant from a fixed line (directrix) and a fixed point (focus) not on the line. 

The point between the midpoint and the focus is called the parabola. Therefore we could calculate the vertex if we knew the focus and directrix. 

The equation of a parabola is:

For a parabola going up the vertices axis- (x-h)^2 = 4p (y-k) 

For a parabola going across the horizontal axis - (y-k)^2 = 4p (x-h) 

The vertex is (h,k) and the focus is always found by using p as well as the directrix.

Here's an example of how to find p.

If we wanted to find the focus, we know that the vertex is (-1,1). Since the parabola is going up, the value of x will remain the same and we will add the value of p to the y of the vertex. The directrix is subtracting it from the vertex.

Monte hall problem

The montee hall show, Let's make a deal, offers an activity that awards prizes. The contestants would choose one of three of doors and one of them would have a prize in it. Once the contestant chose a door, montee hall would open another door that didn't have the prize in it, and he would offer the contestant if he or she wanted to switch doors. The competitor would choose which door from there. 

For several years during that era, it was noticeable that when people switched doors, they got the answer 2/3 of the time while those who didn't got the award 1/3 of the time. Why?

There is actually a bit of math involved. Let's start simple. Picking one door door would give you 1/3 of a chance of getting a prize. Thus you would have 2/3 of a chance picking the wrong door. 1/3+2/3=1. From this you would know that two doors together would have a probability of 2/3 getting the prize. Here's where it gets interesting when he reveals one of the wrong doors. Remember that we already established that the probability of two doors together is 2/3. Staying with your initial choice. Additionally, your initial choice will always be 1/3 of success. If montee hall didn't remove a door, and you changed your mind there are 2 doors to choose from and there is a 2/3 success rate of two doors. Thus, (1/2)(2/3)=2/6=1/3 just like your initial choice. Since he did remove one, you only have one other choice to choose from and remember that two doors has a success rate of 2/3. Therefore 1(2/3)=2/3 probability of success if you switch. Same concept works for four doors. If he didn't remove a door, you have 3 other doors to choose from and a 3/4 success rate of three doors. Thus (1/3)(3/4)=3/12 probability which is just like an initial choice of 1/4 success rate. If he does remove a door there are only two doors to choose from. Thus (1/2)(3/4)=3/8 which is greater than 1/4. Therefore you should always switch doors.

http://montyhallproblem.com

Probability

Probability is defined as

If an event E has n(E) equally likely outcomes and it's sample space S has n(S) equally likely outcomes, the probability is

P(E)=n(E)/n(S)

The probability of an event ranges from 0 _< P(E). _< 1

Must occur at 1 but cannot occur at 0

Let's say we were to determine the the probability of drawing a card less than 6 In a 52 card deck. 

To solve this we know that there are 5 cards less than six and 4 differente suites for each card.

Therefore the n(E) = 5x4 =20

The total sample space is 52.

So P(of drawing a card less than six) = 20/52=5/13= .38

We can also implement count unprincipled in probability

Let's say we were drawing two marbles from a bag of one green, two yellow, and three red marbles. What is the probability of drawing two yellow marbles?

To determine the sample space, we can use combinations to find the number of outcomes

6C2= 6!/(6-2)!2!=15 total outcomes.

The total number of times we can pick two yellow marbles is 1 because there are only two yellow marbles

Thus

P(E)=.07