Line Intercept Sampling

In this article, we will discuss about the line intercept sampling. Sampling means method of selecting sample. The line intercepts have two types, one is x intercept and next one is y intercept. X intercept means that, the point crosses the x -coordinates or axis and y intercept means point crosses the y- axis of the line. The slope intercepts form of line y = mx + b, where m is slope of the line, and b - the y intercept. Let us learn about the line intercept sampling example problems are given below.

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Example problems for line intercept sampling:

Example problem 1:

Find the x intercept of the line, 11.5x + 13.5y = 27

Solution:

The slope intercept form of line y = mx + b, where m is the slope of the line

Given equation is in the form of ax + by = c. To find the x- intercept Plug y = 0 in the equation

Here x intercept, so y = 0

11.5x + 13.5(0) = 27

11.5x = 27

Divide by 11.5 on both sides.

` (11.5x)/(11.5)` = ` (27)/(11.5)`

After simplify this, we get

x = 2.34

x intercept = (2.34, 0)

Example problem 2:

Find the y intercept of the line, 4x + 24y = 72

Solution:

The slope intercept form of line y = mx + b, where m is the slope of the line

Given equation is in the form of ax + by = c. To find the y- intercept Plug x = 0 in the equation

Here y intercept, so x = 0

4(0) + 24y = 72

0 + 24y = 72

24y = 72

Divide by 24 on both sides.

`(24y)/(24)` = ` (72)/(24)`

After simplify this, we get

y = 3

y intercept = (0, 3).

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More example problems for line intercept sampling:

Example problem 3:

Find the y intercept of the line, 5x + 30y = 120

Solution:

The slope intercept form of line y = mx + b, where m is the slope of the line

Given equation is in the form of ax + by = c. To find the y- intercept Plug x = 0 in the equation

Here y intercept, so x = 0

5(0) + 30y = 120

0 + 30y = 120

30y = 120

Divide by 30 on both sides.

`(30y)/(30)` = ` (120)/(30)`

After simplify this, we get

y = 4

y intercept = (0, 4)

The above examples are helpful to learn of line intercept sampling.

Alternating Current Generator

The household electricity that we used is based on the principle of Alternating Current. One advantage of Alternating Current is that it does not induce fatal shock. Also it has an added advantage of easy voltage amplification. In this article we shall discuss the Alternating Current Generator which is often abbreviated as A.C. Generator.

Understanding Alternating Exterior Angles is always challenging for me but thanks to all math help websites to help me out. 

Introduction to A.C. generator

The phenomenon of electromagnetic induction has been technologically exploited in many ways. An exceptionally important application is the generation of alternating current (A.C.).

Principle of ac generator

One method to induce an emf or current in a loop is through a change in the loop orientation or a change in it’s effective area. As the coil rotates in a magnetic field B , the effective area of the loop (the face perpendicular to the field) is AcosӨ, where Ө is the angle between A and B. This method of producing a flux change is the principle of operation of a simple ac generator.

A. C. Generator : Construction and Working

It consists of a coil mounted on a rotor shaft. The axis of rotation of the coil is perpendicular to the direction of the magnetic field. The coil called the armature is mechanically rotated in the uniform magnetic field by some external means . The rotation  of the coil causes the magnetic flux through it to change, so an emf is induced in the coil . The ends of the coil are connected to an external circuit by means of slip rings and brushes.

Working
When the coil is rotated with the constant angular speed of w , the angle Ө between the magnetic field vector B and the area vector A of the coil at any instant t is Ө=wt(assuming Ө=0⁰at t=0). As a result, the effective area of the coil exposed to the magnetic field lines changes with time, and the flux at any time is ф_B=BAcosӨ=BAcoswt From faraday’s law, the induced emf for the rotating coil of N turns is then,  E=-N×dф_B/dt = -NBA×dcoswt∕dt thus, the instantaneous value of the emf is  E= NBAwsinwt.Where NBAw is the maximum value of the emf,which occurs when sinwt= +1 or -1. If we denote NBAw as E₀, then E=E₀sinwt since the value of the sine function varies between +1 or -1, the sign, or polarity of the emf changes with time. 

My forthcoming post is on Perfect Negative Correlation and nmat 2013 syllabus will give you more understanding about Algebra.

The emf has its extremum value when Ө=90⁰ or Ө=270, as the change of flux is greatest at these points. The direction current changes periodically and therefore the current is called alternating current.
The modern ac generator with a typical output capacity of 100 MW is a highly evolved machine.

Polynomials gcf Calculator

Polynomials gcf calculator is one of the interesting topics in mathematics. It is the process of performing greatest common factor for the given polynomial expression. It is the sums of a finite number of monomials are called as polynomial. Polynomial has more than one term and it has a constant value for the given each term, for that variable power of integral is raised to more than two.

Example for Polynomial expression is a2 – 26a – 28.

I like to share this help factoring polynomials with you all through my article.

Definition of Greatest common factor calculator:

Greatest common factor (gcf):

Greatest common factor is defined as the process of the highest number which divides more than two numbers or terms exactly.

Steps to find the Greatest common factor:

Step 1:

Given polynomial expression can be arranged in the order of powers

Step 2:

Each term in the given expression can be factored.

Step 3:

Find the common factors in each terms

Step 4:

Take the greatest common factor

Step 5:

Simplify the each term.

Example problem for polynomials gcf calculator:

Some example problems for polynomials gcf calculator are,

Example 1:

Find the gcf for the given Polynomial expression 3x2 – 9x

Solution:

Step 1:

3x2 – 9x

Step 2:

3 . x . x – 9 . x

Step 3:

In the given expression, common factors in the each term is 3x

3 . x . x – 3 . 3 . x

Step 4:

Take the common term outside

3x ( x – 3)

Step 5:

Solution to the given polynomial gcf is 3x (x – 3)

Example 2:

Find the gcf for the given Polynomial expression 5x2y5 – 20x4y3

Solution:

Step 1:

6x2y5 – 24x4y3

Step 2:

6 . x . x . y . y . y . y . y – 24 . x . x . x .x . y . y . y

Step 3:

In the given expression, common factors in the each term is 6x2y3

6 . x . x . y . y . y – 6 . 4 . x . x . x . x . y . y . y

Step 4:

Take the common term outside

6x2y3 ( y2 – 4x2)

Step 5:

Solution to the given polynomial gcf is 6x2y3 ( y2 – 4x2)

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Example 3:

Find the gcf for the given Polynomial expression 15x3y4 – 45x5y6

Solution:

Step 1:

15x3y4 – 45x5y6

Step 2:

15 . x . x . x . y . y . y . y  – 45 . x . x . x .x . x . y . y . y . y . y . y

Step 3:

In the given expression, common factors in the each term is 15x3y4

15 . x . x . x . y . y . y . y – 15 . 3 . x . x . x . x . x . y . y . y . y . y . y

Step 4:

Take the common term outside

15x3y4 ( y – 3x2  y2)

Step 5:

Solution to the given polynomial gcf is 15x3y4( y – 3x2  y2 )

How to add Trinomials

Trinomials, a function is in the structure of ax2+bx+c =0 (where a≠0, b, c are constants). Quadratic function or quadratic equation also called  trinomials. An algebraic expression which has 3 terms known as trinomials. The product of two binomials gives a trinomial and there are two solutions for the given trinomial. To sum trinomial first it needs to combine the liked terms and then solve it. Since, the trinomial is a combination of two terms and the sum of trinomials also has three terms.

Examples problem for add trinomials:

1. Add trinomials for given function is 2x + 3y + 4z and 6x + 4y +2z.

   Solution:
   Given functions is,
    2x + 3y + 4z and 6x + 4y +2z,

Step 1:
    Write the given trinomials as,
                       = 2x + 3y + 4z + 6x + 4y +2z

Step 2:
    Then combine the terms like as,
                       = 2x + 6x + 3y + 4y + 4z + 2z

Step 3:
    Then add the given trinomials, and we get the answer,
             Answer = 8x +7y + 6z.

2. Add trinomials for given function is 8x - 2y - 9z and 3x + y +2z.

    Solution:
    Given functions is,
     8x - 2y - 9z and 3x + y +2z,

Step 1:
    Write the given trinomials as,
                       = 8x - 2y - 9z + 3x + y +2z

Step 2:
    Then combine the terms like as,
                   = 8x + 3x - 2y + y - 9z + 2z

Step 3:
    Then add the given trinomials, and we get the answer,
            Answer = 11x - y - 7z.

Examples problem for add trinomials:

3. Add trinomials for given function is 3x² + 6y² + 4z² and x² + 5y² + 6z².
    Solution:
    Given functions is,
    3x² + 6y² + 4z² and x² + 5y² + 6z²,
Step 1:
    Write the given trinomials as,
                       = 3x² + 6y² + 4z² + x² + 5y² + 6z²
Step 2:
    Then combine the terms like as,
                       = 3x² + x² + 6y² +5y² + 4z² + 6z².
Step 3:
    Then add the given trinomials, and we get the answer,
            Answer = 4x² + 11y² + 10z².

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4. Add trinomials for given function is x + y + 4z and 6x + 4y +2z.
    Solution:
    Given functions is,
    x + y + 4z and 6x + 4y +2z,
Step 1:
    Write the given trinomials as,
                        = x + y + 4z + 6x + 4y +2z
Step 2:
    Then combine the terms like as,
                        = x + 6x + y + 4y + 4z + 2z
Step 3:
    Then add the given trinomials, and we get the answer,
             Answer = 7x + 5y + 6z.

Sine Cosine Cotangent

Sine:

          A trigonometry functions of an angle. The sine of an angle theta shortened as sin theta In a right angled triangle is the ratio of the side opposite angle to the hypotenuse. This definition applies only of angles between 0 to 90 (0 and `pi/2 ` radians).

Sin `theta` = Opposite / hypotenuse  = `(BC)/(AC)`

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Cosine:

A trigonometry function of an angle.The cosine of an angle `theta` abbreviated as cos `theta` In a right angled triangle is the ratio of the side adjacent to the hypotenuse.

Cos `theta` = Adjacent / hypotenuse = `(BC)/(AC)`

cotangent:

A trigonometry function of an angle.The cotangent of an angle `theta` ( cot  `theta` ) in a right angled triangle is the ratio of the side adjacent to it to the opposite side.
Cot `theta` = adjacent / opposite = `(BC)/(AB)`

Example problem for sin

Find the measure of the length of other sides and also find the sin function values for the given right angle triangle.

we want to find the length of side c, the hypotenuse.

Here, we know that side a has a length of 8 and side b has a length of 6.
To find the length of side c, we can use the Pythagorean Theorem which says that c²=a²+b², or
Substitute in that a=8 and b=6, we find that:

c = √ (( 8²) + (6²))

  = √ (64 + 36)

  = √ 100

c = 10 m

So the value of x is found as x = 10 m
Now we have to find the value of `theta` . we can use the sin function to find the value of `theta`
Sin `theta`     = Opposite / hypotenuse  
          = 6/10
          = 0.6
Sin `theta`  = 0.6
  `theta` = sin^-1 (0.6)
`theta`  = 37<sup>o


Algebra is widely used in day to day activities watch out for my forthcoming posts on Statistics Hypothesis Testing and Integers Number Line. I am sure they will be helpful.</sup>

Example problem for cosine, cotangent:

Find the cosine and cotangent function of the given right angled triangle.

Solution:
Here we have to find the cosine and cotangent of the given right angled triangle
Cosine`theta` = Adjacent / hypotenuse
               =  4 /  5

= 0.8

cos `theta`  = 0.8
`theta` = cos^-1 (0.8)

      = 36^o
Cotangent `theta` = adjacent / opposite =  4/3

= 1.33

cot`theta`  = 1.33
`theta` = cot^-1 (1.33)

Steps in Data Analysis

Analysis of a data is a process of inspecting, cleaning, transforming, and modeling data with the goal of highlighting useful information, suggesting conclusions, and supporting decision making. Data analysis has been the method of multiple facets and approaches, encompassing diverse techniques under a variety of names, in different business, science, and social science domains. And now we see about the steps in data analysis. (Source -Wikipedia).

I like to share this Define Quantitative Data with you all through my article. 

Two tyoes of data analysis are shown below:

• Qualitative Analysis types :

                         The reports are analyzed with the help of given data and that are supported by the qualitative Analysis    .

•  Quantitative Analysis:

                      The reports are  believed in the format of calculating types are called as the  quantitative Analysis . And it functions in the order of  normal information.
Process in data analysis types:
The some of the process in data analysis types are :

• Steps in Data cleaning:

       Data cleaning types are the is a main types for the interval of in which the data are experienced, and False  data are if important, preferable, and realistic corrected.

• Steps in Preliminary Data analysis :

       Steps in  preliminary data analysis are,

• Steps  in Quality of Data: Data can be used in a   set of  types,

         Like  Histogram, Standard possibility in plotting mean,Average variation,Median

• Steps  in Quality  analysis of size in methods:

                   The assessment of homogeneity, which provide a deference of the reliability of a component tool 

• Main steps in  data analysis:

       The main difference between of  the preliminary data analysis and the main analysis are they can get the information’s side by side.

• Types in final data analysis :

     During the final stage of  period   , the requirements  of the  of the preliminary data analysis are predictable.


Algebra is widely used in day to day activities watch out for my forthcoming posts on Simplify Using Positive Exponents and Mixed Fraction to Decimal. I am sure they will be helpful.

Example in Data analysis steps:

Example 1:
Represent the following data  in statistical histograms given below
Marks                             No.of students
0 - 5                                        2
5 - 10                                      3
10 - 15                                    4
15 - 20                                    6
20 - 25                                    2
 Solution:
Step 1: To mark the class intervals along the x – axis.
              Student marks intervals can be , 0-5, 5-10, 10-15, 15-20, and 20-25.
Step 2: To mark the frequencies along the y – axis.
               Frequencies for number of students are,4,2 , 4, 6 and 2.
               Therefore, the statistical histograms learning to depict the scale in the X-Y plane :
                                    x - axis 1 cm = 5 marks.
                                   y - axis 1 cm = 1 students
Step 3:. Total frequency = 2 + 3 + 4 + 6 + 2 = 17
           And the bar graph are drawn below:

                                

Inverse Correlation Definition

Correlation is a statistical tool which measures the degree and the direction of relationship between two or more variables.Thus correlation is a statistical device which helps us in analysing the covariation of two or more variables.

Types of Correlation

Correlation is described or classified in several different ways. Three of the most important ways of classifying correlation are:
(i) Positive or Negative(Inverse)
(ii)Simple, partial and multiple.
(iii)Linear and non-linear.

Positive and Negative Correlation.
Whether correlation is positive(direct) or negative(inverse) would depend upon the direction of change of the variables. If both the variables are varying in the same direction, i.e., if as one variable is increasing the other on an average is also increasing or, if as one variable is decreasing the other on an average is also decreasing, correlation said to be positive.If, on the other hand, the variables are varying in opposite directions, i.e., as one variable is increasing the other is decreasing or vica-versa, correlation is said to be negative(Inverse).
In other words, when two variables move in the same direction they are said to be positively correlated and when two variables moves in the opposite direction they are said to be negitively correlated.
The following examples would illustrate the difference between positive and negative correlation.

1. POSITIVE CORRELATION

X     :           10        12        15        18        20               X:          80         70         60         40         30
Y     :           15        20        22        25        37               Y:          50         45         30         20         10
2.  NEGATIVE(INVERSE) CORRELATION
X     :           20        30        40        60        80               X:         100         90         60         40         30
Y     :           40        30        22        15        10               Y:           10         20         30         40         50

Inverse correlation

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In a inverse correlation, as the values of one of the variables increase, the values of the second variable decrease. Likewise, as the value of one of the variables decreases, the value of the other variable increases.

This is still a correlation. It is like an “inverse” correlation. The word “negative” is a label that shows the direction of the correlation.

 Here are some other examples of negative correlations:

1. Education and years in jail—people who have more years of education tend to have fewer years in jail (or phrased as people with more years in jail tend to have fewer years of education)

2. Crying and being held—among babies, those who are held more tend to cry less (or phrased as babies who are held less tend to cry more)