Height of Trapezoid

Geometry deals with shapes, structures, lines, planes and angle’s. Geometry learning is also known as architectural learning. Basic shapes of geometry are square, triangle, rectangle, parallelogram, trapezoid etc. Trapezoid is one of the basic shapes in geometry. Trapezoid is a quadrilateral which has 4 sides. The total internal angle of the trapezoid is 360 degree. In trapezoid, one pair of opposite sides is parallel.

Formula for finding the height of the Trapezoid

The formula for finding the area of pyramid is given as,

Area of pyramid = h (b1 + b2)/2

Where,

h = height of the pyramid,

b1, b2 = bases of the trapezoid.

From the given area formula we can find the height of the trapezoid when the area of trapezoid is given,

A = h (b1 + b2)/2

2A = h (b1 + b2)

h = 2A/b1+ b2



Problems on height of pyramid:

Example 1:

Find the area of the trapezoid, whose bases are 10 cm and 12 cm, height, is 6 cm.

Solution:

Formula for finding the area of the trapezoid is,

Area of pyramid = h (b1 + b2)/2

= 6 (10 + 12) / 2

= 6 (22) / 2

= 3 * 22

= 66 cm2.

The answer is 66cm2.



Example 2:

Find the height of the trapezoid, whose bases are 8 cm and 12 cm, area, is 120 cm2.

Solution:

Formula for finding the area of the trapezoid is,

Area of pyramid = h (b1 + b2)/2

120 = h (8 + 12)/2

120 * 2 = h (20)

240 =   h*20

Divide 20 on both sides,

240/20 = 20*h/20

12 = h

The height is 12 cm.

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Example 3:

Find the height of the trapezoid, whose bases are 6 cm and 4 cm, area, is 100 cm2.

Solution:

Formula for finding the area of the trapezoid is,

Area of pyramid = h (b1 + b2)/2

100 = h (6 + 4)/2

100 * 2 = h (10)

200 =   h*10

Divide 10 on both sides,

200/10 = 10*h/10

20 = h

The height is 20 cm.

Least Common Multiple of 3 and 6

In mathematics, the least common multiple of two rational numbers a and b is the smallest positive rational number that is an integer multiple of both a and b. Since it is a multiple, it can be divided by a and b without a remainder. If either a or b is 0, so that there is no such positive integer, then LCM(a, b) is defined to be zero. (Source: From Wikipedia).

Least common multiple of two numbers can be found by their multiples. Here we are going to learn how to find the least common multiple of two or more numbers.

Least common multiple of 3 and 6

The least common multiple of 3 and 6 can be found by finding the multiples 3 and 6.

The list of multiples of 3 and 6 are given below

The multiples of 3 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60.

The multiples of 6 = 6, 12, 18, 24, 30, 36, 42, 48, 54, 60.

Here 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 are the common factors, among those 6 is the lowest common number.

So, 6 is the lowest common multiple of 3 and 6.

Example problems for least common multiple

Example 1

Find the least common multiple of 3 and 16

Solution

The least common multiple of 3 and 16 can be found by finding the multiples 3 and 16.

The list of multiples of 3 and 16 are given below

The multiples of 3 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60.

The multiples of 16 = 16, 32, 48, 64, 80, 96, 112, 128, 144, 160.

Here 48 is the lowest common number. So, 48 is the lowest common multiple of 3 and 16.

Example 2

Find the least common multiple of 13 and 6

Solution

The least common multiple of 13 and 6 can be found by finding the multiples 13 and 6.

The list of multiples of 13 and 6 are given below

The multiples of 13 = 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143.

The multiples of 6 = 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 72, 78, 84.

Here 78 is the lowest common number. So, 78 is the lowest common multiple of 13 and 6.

My forthcoming post is on Positive Correlation Graph and percentage formulas will give you more understanding about Algebra.

Example 3

Find the least common multiple of 13 and 16

Solution

The least common multiple of 13 and 16 can be found by finding the multiples 13 and 16.

The list of multiples of 13 and 16 are given below

The multiples of 13 = 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 156, 169, 182, 195, 208.

The multiples of 16 = 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208.

Here 208 is the lowest common number. So, 208 is the lowest common multiple of 13 and 16.

Beginner Multiplication

Multiplication (x) is the arithmetical operation of calculating one value by another value. It is a one kind of essential operations in basic arithmetic (the others operations are addition, subtraction and division). Since the outcome of calculating by whole numbers can be thinking of as including of a few number of copies of the original, whole-number products larger than 1 can be calculated by frequent addition.

Types of multiplication for beginner:

• Multiplication of variable by exponent

• Multiplication of fraction

• Multiplication of unlike signs ( positive(+),negative (-))


Understanding Multiplication Fractions is always challenging for me but thanks to all math help websites to help me out.

Multiplying Variables with Exponents:

Multiplication of exponent:

1) Exponent of 0:

If the exponent is 0 specifies you are not multiplying by anything and the answer is 1

For example, a0 = 1

x0 = 1

2) Exponent of 1:

If the exponent is 1 specifies you are multiplying the variable with 1. (Example x1 = x)

Rules for multiplication terms on fractions:

• First multiple the value of numerator.

• Then multiple the value of denominator.

• Lastly decrease the fraction (if required).

Example:

`2/5` ×` 3/4`

Step1 Multiply the numerators:

`3 / 4` × `2 / 5` = > 3 × 2 / 4 x 5 = `6/20`

Step2 Multiply the denominators:

` 3/4` ×`2/5` = 3×2 / 4 x 5 = `6/20`

Step3 Lastly decrease the fraction

Therefore solution is `3/10` .

My forthcoming post is on Population Versus Sample and The Prime Numbers will give you more understanding about Algebra.

Multiplication of unlike signs for beginner:

• Positive(+) × Positive(+) = Positive(+)

Ex:  6 × 2   =12
• Positive (+) × Negative (-) = Negative (-).

Ex:   6 × (-2) = -12
• Negative (-) × Positive (+) = Negative (-).

Ex: (-6) × 2 = -12
• Negative(-) × Negative(-) = Positive(+).

Ex: (-6) × (-2)=12

Example:

Multiply by (a + 4) (a -5)

Step 1: multiply by y in the second factor

a (a-5) =  a2- 5a

Step 2: multiply by 5 in the second factor

4(a - 5) = 4a-20

Step 3: add step 1 and step 2

(a + 4) (a - 5) = a2- 5a + 4a - 20

= a2 -1a - 20

Basic multiplication problems for beginner:

1) Multiply the values i) 23 × 2 ii) 4 × (6 × 3) = (4 × 6) ×3

Solution:

i) 23 × 2 = 46

ii)    4 × (6 × 3) = (4 × 6) ×3

4 × 18 = 24 × 3

72 = 72.

2) Multiply the values i) 4 × (2 + 6)   ii) 6 × (5 × 3) = (6 × 5)×3

Solution:

i) 4 × (2 + 6) = 4 × 2 + 4 × 6

= 8 + 24

= 32.

ii)6 × (5 × 3) = (6 × 5) × 3

6 × 15 = 30 × 3

90 = 90.

Isosceles Triangle Hypotenuse

When the two sides of the triangle are said to be equal then the triangle is called as isosceles triangle. When all the sides of the triangle are equal then the triangle is called as the equilateral triangle and when no sides of the triangle are equal then it is said to be scalene triangle. In an isosceles triangle only the sides but also the two angles are said to be equal.
                                                               
Here we will see about the isosceles triangle hypotenuse.

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Isosceles triangle hypotenuse.

The hypotenuse side of the isosceles triangle is the sides that are found opposite to that of the right angle. According to the right angle theorem the height of the isosceles triangle is given by
                         h = √(b² – ¼ a²)
Thus the area of an isosceles triangle is
                        A = ½ ah
                            = ½ a √(b² – ¼ a²)
                            = ½ a² √((b²/a²) – (¼))
An isosceles triangle is also called as the triangle with two congruent sides. The angles which are opposite to these congruent sides are called as the base angles and the angles found between those sides are called as the vertex angle of the isosceles triangle.
Any equilateral triangle can be an isosceles triangle but no isosceles triangle is an equilateral triangle.                           

Properties of the isosceles triangle base:

• According to the right angle theorem the sum of the squares of the hypotenuse side is equal to the sum of the squares of the other two sides. The hypotenuse side is the side which is opposite to the right angle

• The side which is not equal to the other sides of a triangle is called as the base of the isosceles triangle.

• The base angles of the isosceles triangle are found to be equal.

• When the third angle of the isosceles triangle is a right angle then it is called as the right isosceles triangle.

• The perpendicular distance from the base to the vertex of an isosceles triangle is called as the altitude of the isosceles triangle.

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Externally Tangent Circles

Two intersect circles in a single point is known as tangent circles. It can be divided into two types of tangency: internal and external. By using the tangent circles many problems and constructions in geometry are solved. This type of problems have real-life applications. The followings are some real-time applications: trilateration and maximizing the use of materials.

                                             

I like to share this Tangent geometry with you all through my article. 

Illustration:

A tangent that is common to two circles and does not intersect the segment joining the centers of the circles is called Common External Tangent. A common tangent can be any one of the following:External tangent or Internal tangent.

                                                   
          From the above figure, we can see that line PQ is the common external tangent.

                                               
          From the above figure, we can see that line AB is the common external tangent.

Theorem for Externally Tangent Circles:

Theorem:If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of tangency.
                                                       
Solution:
            If:        Line  AB is a tangent
                       D is point of tangent.
            Then:   OD ┴ AB

Example problems:

Example 1: Find the common external tangent from the below figure.

                                             
Solution:
            A tangent that is common to two circles and does not intersect the segment joining the centers of the circles is called common external tangent.
            From the given figure, the line CD touches both the circles on the same side of the line is the external tangent to the circle.
            Therefore, CD is the common external tangent line.

Example 2: Find the common external tangent from the below figure.
                                                                                    
Solution:
            The given figure is concentric circles. Concentric circles having the same center and not tangent.

Example 3: Find the common external tangent from the below figure.
                                                     
Solution:
          From the above figure, we can identify that given is two complete separate circles.
          The given figure have two external tangents.

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Example 4: Find the equation for tangent lines y=3x³ from the point x= 2.
Solution :
Given, y=3x³………….(1)
Step 1: To find the first derivative of y

                y ' = 3x ² ………..(2)
Step  2: To plug x =2 into y ' to find the slope at x.
                y ' = 3(2) ²
                y= 12
               Slope of tangent line =12
Step 3:To plug x=2 into y to find the y value of the tangent point
              y =3(2)³
              y=24 
              Hence tangent point (2,24)
Step 4: To plug the slope=12 and point (2,24) using the point-slope equation to find the equation     for the tangent line.
              y- y₁ =m (x-x₁)
              y- 24 =12 (x-2)
              y =12x- 24 + 24
              y=12x
Example 5: Find the slope of tangent line which is passing through the point (2, 0) and the curve y = 2x – x³
Solution:
            Step 1: Find the derivative y¹
                                y = 2x – x³
                                y¹ = 2 -3x²
            Step 2: To find the slope of tangent line,
                         Given point (2, 0) --- > (x, y)
                         y¹ = 2 -3(2)² ---- > 2 – 3(4) --- > 2 -12 = -10
                         Slope of the tangent line = -10         

Practice problems:

1. Determine the equation of the tangent line passing through the point (− 1, − 2) and having slope4 /7
Answer:   4x − 7y − 10 = 0
2. Determine the equation of the line with the slope 3 and y-intercept 4.
Answer: y = 3x + 4

Power Factor Conversion

To signify a lot of the numbers we have used scientific notation. For example the number 7500 is identified as 7.5E3. Here the letter “E" can be read as "times ten to the power of". Every unit is written out in full before its contraction is used. Some units have superscripts in them. In power factor conversion includes the watt to joule, foot pound to force per second, kilogram to meter per second, horse power to watt.

I like to share this Greatest Common Factor Examples with you all through my article.

Example problems of power factor conversion:

Power factor conversion Problem 1:

Convert watt to joule per second.

25 W

30 W

Solution:

We can convert watt to joules per second by using the following method.

1 watt = 1 joule per second

The initial unit is 25 W and 30 W

Here conversion factor is 1 W/ 1 joule/sec

That is, `25*` `1/1` = 25

`30*` `1/1` =30

Therefore the final unit is 25 and 30 units

Answer: 25 joule/sec and 30 joule/sec

Power factor conversion Problem 2:

Convert watt to joule per minute.

55

Solution:

We can convert watt to joule per minute by using the following method.

1 watt = 60 joule per minute

The initial unit is 55 W

Here conversion factor is 1 watt = 60 joule per minute.

That is, `55*` `60/1` =3300

Therefore the final unit is 3300 units

Answer: 3300 joule/minute

Power factor conversion Problem 3:

Convert watt to horsepower.

25

Solution:

We can convert watt to horsepower by using the following method.

1 watt = 0.001341022 horsepower

The initial unit is 25 watt

Here conversion factor is `(0.001341022 hp)/(1 watt)`

That is, 25*0.001341022= 0.033525552 horsepower

Therefore the final unit is 0.033525552 horsepower

Answer: 0.033525552 horsepower

Practice problems of power factor conversion:

1. Convert watt to joule per second: 75

2. Convert watt to joule per minute: 22

3. Convert watt to horsepower: 45

My forthcoming post is on Solve Equation Online and math 3rd grade word problems will give you more understanding about Algebra.

Answer:

75 joule per second
1320 joule per minute

3.  0.060345994 horsepower

Prime and Rate

Prime: Definition of prime number is a number which cannot divide by any other numbers. That is ,only the number itself and 1 can divide the prime number.
        Rate: Definition of rate is the ratio between the distance and time. Definition of rate is always in the form of fraction.
Let us see the definition of prime and rate.

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Definition of prime number:

Definition of prime number:

If a number is not allowed to divide by any number then it is called as prime number. Other wise it is called as composite number.

Example: 133

This number can not be divide by any number. Only 1 and 133 can divide it. So, 133 is considered as prime number.

Example:1

Find the prime number among the following numbers.

a. 155

b. 242

c. 141

d. 339

Ans :141

Example: 2

Find the prime number among the following numbers.

 a.122

b.223

c.144

d.166

Ans : 223

Definition of rate:

Definition of rate:
  Formula for finding a rate is,

Rate=Distance/Time

        Rate has a units depends on the units of distance and time. If the distance has a unit in km and the time is in hr then the rate should be written as km/hr.
Example problems for finding a rate:
Example:1

Find the rate of scooter. The distance travelled by a scooter is 89 km and the time taken for that is 3 hours.

Solution:
Given,

Distance = 89km

Time=3 hr

We know the rate formula is Rate=distance/time

Rate=`89/3`km/hr.

Therefore the rate of the given scooter is `89/3` km/hr.

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Example:2

Find the time taken by a cycle. The distance travelled by a cycle is 24 km and the rate is `12/6`km/hr.

Solution:
Given,

Distance = 24km

Rate=`12/6`km/hr

We know the rate formula is Rate=distance/time

Time= Distance/time

         =`24/(12/6)`

        =`144/12`

       =12 hours.

Therefore the time taken by a cycle to travel 24 km is 12 hours.

Fraction Conversion Table

Fraction:
A fraction is a number that can represent part of a whole. The earliest fractions are reciprocals of integers: ancient symbols represent one part of two, one part of three, one part of four, and so on. A much later development was the common fractions which are used today (½, ⅝, ¾, etc.) and which consist of a numerator and a denominator.

Fraction can be converted into to forms as shown below,

• Fraction conversion into decimal
• Fraction conversion into percentage.

Through this article you can learn about fraction conversion table, how to make conversion of fraction into decimal and how to make conversion of fraction into percentage.

I like to share this Fraction Equivalents with you all through my article. 

Example for fraction conversion table:

Fraction conversion table:

         We can express the conversion of fraction in terms of table format.This table is known as fraction conversion table.The following are the two examples for fraction conversion table.

                       

Fig(i) Fraction conversion table 1                                      Fig(ii) Fraction conversion table 2

The fractions that contains the denominator as a  power of  10 called as decimal fractions.
Ex:
0.1 = 1/10
0.01 = 1/100
0.001 = 1/1000

Problems on fraction conversion table:

Pro 1: Convert the fraction 45 / 90
Sol:
Given, 45 / 90 into decimal.
To find the decimal,divide the numerator by the denominator ,
             _________
         90 )  45      (
45 cannot be divided by 90.
So Multiply 45 by 10 and add one decimal place in quotient,
             _________
         90 )  450     ( 0.5
                450     
                 0        
So the decimal of fraction 45/90 is 0.5
Ans: 45/90 = 0.5
Pro 2: Convert the fraction 76 / 43 into decimal.
Sol:
Given, 76 / 43 as decimal
To find the decimal,divide the numerator by the denominator ,
             _________
         43)  76     ( 1.76
               43     
                33      
33 cannot be divide by 43. So multiply 33 by 10 and one decimal place in quotient,
            _________
         43)  76     ( 1.7
               43     
               330    
               301   
                 290
                 258
                   38
The decimal for 76 / 43 = 1.76
Ans:76 / 43 = 1.76

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Pro 3: Convert the fraction 6 / 5 into decimal.
Sol:
Given, 6 / 5 as decimal
To find the decimal,divide the numerator by the denominator ,
           _________
        5 )  6     ( 1
              5     
              1   
1 cannot be divide by 5. So multiply 1 by 10 and one decimal place in quotient,
            _________
           5 )  6     ( 1.2
                 5     
                 10
                 10  
                   0 
The decimal for 6/5 = 1.2
Ans: 6/5 = 1.2

Venn Diagram Example of Disjoint

Let use see about venn diagram example of disjoint.Venn diagrams also called as the set diagram.These are diagrams that show all theoretically possible logical relations between finite groups of set. Venn diagrams were considered approximately 1880 by John Venn. They are used to teach elementary set theory, and show simple set relationships in logic, probability and statistics.


venn diagram example of disjoint - Notations


Curly  braces - {...} - are used to phrase.
These braces can be used in various ways.
For example:

• List the elements of a set: {-3, -2, -1, 0, 1, 2, 3,4}

• Describe the elements of a set: {integers between -3 and 3 inclusive}

• Use an identifier (the letter x for example) to symbolize a typical element, a '|' symbol to stand for the axiom such that', and then the rule or rules that the identifier must follow: {x | x is an integer and |x| < 5}

The Greek letter ∈ is used as follows:

• ∈ means 'is an element of ...'. For example: 3 ∈ {positive integer}

• ∉ means 'is not an element of ...'. For example: Washington DC ∉ {European capital cities}

• The set is a finite: {British citizens}

                      Or

•  infinite: {6, 12, 21, 24, 35, ...}

Sets are usually be represented using upper case letters: A, B,X,Z ...

venn diagram example of disjoint - Example

The following is the diagram representation of disjoint in venn diagram.
Two sets are equally exclusive also called disjoint. If do not have any elements in common and need not together contain the universal set.
The following venn diagram represents the disjoint sets.

Set of all elements of A is also known as difference of set A-B, which do not go to B. In the set planner form, the difference set is :   

 

Example problem for disjoint set.
A={2,3,4,1,8,9}   B={2,3,4,1,10,12} What is the A-B and B-A?
Solution:
A-B=?
Given A={2,3,4,1,8,9}
            B={2,3,4,1,9,10,12}
Here all elements of A  an available in B except 9.
So the A-B is 9.
B-A=?
Here all elements of A  an available in A except 12.
So the B-A is 12.

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).

My forthcoming post is on Laws of Logarithms and iit entrance exam 2013 will give you more understanding about Algebra.

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


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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)