Ray Lewis vs a batering Ram

Hello mathland, I know it has been a while since the football season ended, but I am doing another post about football. In this sports science video, we see the general idea of how hard a NFL lineback hits as we compare it with a battering ram. This is overall a generally interesting video to watch, and we also learn how much force it takes to break through a door.
https://www.youtube.com/watch?v=k8krW5z-MO4

Counting principles

Fundamental counting principle

E1 and E2 are two different events. The first event can happen in m1 different ways and the second event can happen in m2 different ways. M1 x m2 is the number ways that two events can occur. 

For example if we were to find the total different combinations of telephone numbers for each area code, we see that there are eight different events. (Important to note that the first digit cannot be 0 or 1, thus giving us eight ways event one can occur). 

8 x 10 x 10 x10x10x10x10. = 8,000,000. 

Permutation (order matters)

A permutation of x number of elements is an ordering of elements. 

For instance, if 4 people were in a line, we could be asked how many different ways could they line up.

In this case order matters.

4 different people can take the first spot, then 3 different people can take the second part since somebody already took the first spot. And so on.

4! = 24 different ways. 

Let's also say that 8 track runners ran in a race. How many different combinations of runners can get 1st, second, and third.

In this instance, we use the equation nPr= n!/(n-r)!

Thus 8P3=8!/(8-3)! =336

Combinations (order doesn't matter)

The number of combinations taken r at a time is denoted by 

nCr=n!/(n-r)!r!

Let's say we were choosing 4 marbles from a bag of 7 marbles. How many different combinations of 4 can we get?

7C4=7!/(7-4)!4! = 35 different ways.

Binomial expansion

In a binomial raised to a certain power, it can be difficult to calculate the coefficients and exponents of the seperate terms. However there are patterns that are noticeable when expanding a binomial.

1. In each expansion, there are n+1 terms.

2. X and y symmetric roles. The powers of x decrease while the powers of y increases by 1 successively.

3. The sum of powers of each term is n. 

4. Coefficients increase and decrease

Binomial coefficients 

nCr = n!/(n-r)!r! 

r is the specific term in the expansion. The fourth term would be r =3 which means the number of term is r+1. 

For instance...

The third term for (x+y)^4 is 6

4C2 = 4!/(4-2)!2! =6

The pascal Triangle is also a means of determining the coefficients since it already states the coefficients for an (x+y)^n. 

Here's an example of an expansion...

Blog 20: mathematical induction

One of the only reasons why the mathematical induction works is beacause of the well ordering principle. The well ordering principle states that every non empty su stet of a natural number has a least element. We use mathematical induction in order to prove an equation. 

Let's say we were tryin to prove the equation that

(2n-1)=n^2

Step 1: we have to prove the statement is true at the starting point (n=1)

Step 2: assume tha the statement is true for n. Prove the statement is true for n+1

Enginerring in Half-Pipe

Hey yall, since its almost the end of the Sochi WInter Olympics, I found it appropriate to do this math blog on one of the events. My favorite event is the snowboard half-pipe, which involves performing several airborne tricks on a litteral snow half-pipe. However, in order for one to perform such daring tricks, the half-pipe also must be designed in a certain way. One factor for a snowboard is acceleration. This is defined as the rate of change in velocity. Therefore, if one wanted to gain more height on the halfpipe, that individual would have to increase his acceleration, since it gives a greater force on the body. Gravity also contributes to the performance. The walls of a half pipe are 22 feet and the width is approximately 65 feet wide. This fits a greater radius with the pipe, and thus greater room for speed.
works cited
http://www.nsf.gov/news/special_reports/winterolympics/halfpipe.jsp

Post 19: arithmetic sequences

Arithmetic: a sequence whose common terms have a common difference 

For instance

7,11,14,18

There is a difference of four between each term

We can find the An term of a sequence in the equation

An=A1+(n-1)d

D is the common difference

Arithmetic sequence

Sn=n/2(A1+An)

This is the sum of a finite sequence

Say we were given the sequence

1,3,5,7

Sn=(4/2)(1+7) = 16

Post 18: sequences and summation notation

How do we find the terms of a sequence?

Say we were the sequence An = 3n - 2

A1 = 3(1) - 2=1

A2 =3(2)-2=4

....

We must substitute the variable n with the specific term number to find the value of that term.

A recursive sequence

In order to find the terms we have to be given the value of a few terms and also be given the recursive formula

For example

A0=1,A1=1 Ak=Ak-2+Ak-1

 This means that one term is equivalent to the sum of the two before it. 

Therefore

A2=A1+A0=1+1=2

A3=A2+A1=2+1=3

Summation notation

This involved the uppercase of Greek Letter sigma 

Basically it gives the number of terms in a sequence and the pattern. After finding the value for each term, we add them together. We are also given the term we start on.

Post #17 a review of matrices

Here's a link to a google site that reviews matrices

https://sites.google.com/a/maranathastudents.org/group-5/home

blog #15: using matrices to find an equation of the line

February 7, 2014
here is a link for solving the equation of a line using matrices.

Post #13- sample problems on determinants

So basically I solved the 2x2 matrices by first multiplying the left corner by the bottom right corner. I subtracted this by the product of the bottom left on the top right corner. That's the determinant for a 2x2

Post #14: encryption matrices

So apparently in math land we can actually create codes using the codes. They are very hard, but it can be very interesting to see what kind of message can be created. 

The way to do this is to start creating the inverse of your matrix (decryption matrix). After we find this inverse (say mxm), we will multiply this by another matrix (1xm). We will continue to multiply the original matrix by as many types of 1xm matrices depending on the length of the code. From the results we have, we will discover the code. We also have a scale that translates the numbers into letters. For example 0=space, 1= a, 2=b..... 

Post #12 inverse matrices

Here are two sample problems that I did in order to find the inverse matrix.

So basically in problem number one there was no inverse. I was able to figure this out because the third row only contained zeroes which meant I could go no further. 

Remember to also apply the elementary row operations for the other matrix as well. 

Remember that A x the inverse or A is always the leading one matrix.