Multiple Determinism

 Determinism is the philosophical view that every event, including human cognition, behavior, decision, and action, is casually determined by the environment. It is, in essence, the view that one's life is predetermined before one is even born. Determinism proposes there is a predetermined unbroken chain of prior occurrences back to the origin of the universe.

Determinists believe the universe is fully governed by causal laws resulting in only one possible state at any point in time. With numerous historical debates, many varieties and philosophical positions on the subject of determinism exist, most prominently the free will disputes  involving compatibilism and incompatibilism.

In other words, determinism necessarily entails that humanity or individual humans may not change the course of the future and its events (a position known as fatalism).However, some theorists  believe that the extent to which human beings  have influence over their future is itself merely dependent on present and past. Causal determinism is associated with, and relies upon, the ideas of materialism

Multiple determinism

The Principle of Multiple Determinism

The principle of multiple determinism specifes that a

target event at one level of organization, but particularly

at molar (e.g., social) levels of organization,

may have multiple antecedents within or across levels

of organization. For example, consider the multiple

factors that contribute to drug abuse.

Onthe biological

level, researchers identi®ed the contribution of individual

differences in the susceptibility of the endogenous

opiod receptor system while on the social

level investigators have noted the important role of

social context. 

Both operate, and our understanding

of drug abuse is incomplete if either perspective is

excluded. A corollary to this principle is that the

mapping between elements across levels of organization

becomes more complex (e.g., many-to-many) as

the number of intervening levels of organization

increases. One implication is that the likelihood of

complex and potentially obscure mappings increases

as one skips levels of organizations.

Preparation for Inverse Exponential Function

The exponential function is the function e^x, where e is the number (approximately 2.718281828) such that the function e^x equals its own derivative. The exponential function is used to model phenomena when a constant change in the independent variable gives the same proportional change (increase or decrease) in the dependent variable. (Source from Wikipedia).It is otherwise Written as exp(x).Here we are going to see about preparation about inverse exponential function and its example problems.

Preparation for finding the inverse of the function:

• First we got to to replace f(x) with y
• Switch x‘s   y‘s
• Solve for y
• Replace y of f ^–(x)                   

Preparation problems for preparation for inverse exponential function:

       pro1 :  Find the inverse function of the following y = 3x+1

Sol :  The given function is y = 3x +1

Following the above procdure to find the inverse of the function

First we have to Interchange x and y value

x=3y+1

Then now solve for y

Add both sides -1 then the equation will be

x-1 =3y + 1 -1

x-1=3y

Divide both sides 3 we get ,

y = `(x-1) / (3)`

Therefore the inverse function is  (x-1) / (3)

Example for inverse of exponential function:

        Pro 2:  Find the inverse of the following exponential function,
                    y = 5. 6 ^(3x-4) +7

Sol :   Interchange the x and y value then the equation will be,

x= 5. 6 ^(3y-4) +7

Now we solve for y

Add both sides -7 we get

x-7 = 5. 6 ^(3y-4) +7-7

x-7 = 5. 6 ^(3y-4)

Divide both sides 5 we get,

`(x-7) / (5)` = `(5. 6 (3y-4)) / (5)`

`(x-7) / (5)` = 6 ^(3y-4)

Take both sides log then the equation will be ,

Log ₆ `((x-7) / (5))` = log ₆ 6 ^(3y-4)

Log ₆ `((x-7) / (5))` = 3y-4

Add both sides +4

Log ₆ `((x-7) / (5))` + 4 = 3y

Both sides divide 3 we get

`{Log 6 ((x-7) / (5) + (4))} / (3)` = `(3y) / (3)`

`{Log 6 ((x-7) / (5) + (4))} / (3)` = y

Therefore the inverse exponential function is `{Log 6 ((x-7) / (5) + (4))} / (3)` = y

Hence the preparation of inverse exponential function is explained,

Figure Segment of a Circle

In geometry, a circular segment also circle segment. It is an area of a circle informally defined as an area which is "cut off" from the rest of the circle by a secant or a chord. The circle segment constitutes the part between the secant and an arc, excluding the circle's center.It is shown in figure bellow.
  

Understanding Circle Formulas is always challenging for me but thanks to all math help websites to help me out. 

Example of figure segment of a circle:

Example 1
Radius of the circle is 15 cm, angle of the Segment of a circle is 45 degree. Find the area of the Segment of a circle is shown in figure.


Solution:
Area of the circle = `1/2` r²`theta` - sin `theta`

                            = `1/2` r²( `pi/180` *`theta` - sin `theta`)                       

                             =0.5*15²(`pi/180` *45-(sin 45))

                             =112.5(0.785-0.7071)

   Area of the Segment of a circle=8.77 cm²

Example 2
          Radius of the circle is 10 cm, angle of the Segment of a circle is 55 degree. Find the area of the Segment of a circle is shown in figure.
 
Solution:

Area of the circle = `1/2` r²`theta` - sin `theta`

                            = `1/2` r²( `pi/180` *`theta` - sin `theta`)                       

                             =0.5*10²(`pi/180` *50-(sin 50))

                             =50(0.873-0.766)

   Area of the Segment of a circle=5.53 cm²

Example 3
Radius of the circle is 15 cm, angle of the Segment of a circle is 85 degree. Find the area of the Segment of a circle is shown in figure.

Solution:
Area of the circle = `1/2` r²`theta` - sin `theta`

                            = `1/2` r²( `pi/180` *`theta` - sin `theta`)                       

                             =0.5*15²(`pi/180` *85-(sin 85))

                             =112.5(1.484-0.996)

   Area of the Segment of a circle=54.9 cm²

Example 4
Radius of the circle is 15 cm, angle of the Segment of a circle is 105 degree. Find the area of the Segment of a circle is shown in figure.
Solution:
Area of the circle = `1/2` r²`theta` - sin `theta`

                            = `1/2` r²( `pi/180` *`theta` - sin `theta`)                       

                             =0.5*15²(`pi/180` *105-(sin 105))

                             =112.5(1.833-0.9659)

   Area of the Segment of a circle= 97.54 cm²

Example 5
Radius of the circle is 15 cm, angle of the Segment of a circle is 110 degree. Find the area of the Segment of a circle is shown in figure.

Solution:
Area of the circle = `1/2` r²`theta` - sin `theta`

                            = `1/2` r²( `pi/180` *`theta` - sin `theta`)                       

                             =0.5*15²(`pi/180` *110-(sin 110))

                             =112.5(1.92-0.9396)

Area of the Segment of a circle=110.29 cm²

Distribution of the Sample Means

A sample mean is a numerical set which is an average value of a particular portion of a number in a certain group. A sample mean is expressed as x.suppose if we take a sample of size n and there is a n independent variables x1,x2…xn and each value is respect to one randomly selected observation, then the distribution of the population  of these variables has the mean value mhu and the standard deviation sigma. Then the sample value is,  
                             X=1/n(x₁+x₂+…..x_n)

Mean and variance of the distribution of the sample means:

Using the mean and variance property of the random variable ,the mean and variance of the sample mean is given below.
µ_x= µ
sigmax=sigma/sqrt(n)`
therefore the mean value of the sample mean is similar to the mean of the population distribution and the variance is smaller than the variance of the population distribution.

For example:
In population distribution  where the mean value µ=20 and the standard deviation σ=2 that means ( N(20,2).then we have the simple random sample of 100 students ,then what is the  mean and variance of  the sample mean distribution?
                                                µ_x= µ
                                                µ_x=20
                                                σ_x=σ/`sqrt(n)`
                                                σ_x =2/`sqrt(100)`
                                                σ_x =2/10
                                                 σ_x =0.2

shape property of the distribution of the sample means:

It may be in the following shapes like normal, skewed, bimodal.

Explanation for distribution of the sample means:
If the population distribution is normal,then the distribution of  the sample mean is normal. If in the population distribution   the mean and variance value is respectively  (µ, σ), then the mean and variance for the distribution of the sample mean is(µ, σ/`sqrt(n)` ).
 For linearcombination the sample mean x=(1/n)(x₁+x₂+..x _n)
By using the above all concept the following theorem is illustrated.

Central Limit Theorem- distribution of the sample mean:
Theorem definition:

If we take a  population with a  mean μ and a variance σ², then the  sampling distribution of the mean approach a normal distribution with a mean of μ and a variance of σ²/N as N means   increases value of the sample size.

Irregular Triangle

A closed figure contains the three line of the segments that join end to end. The triangle is the three side polygon. The irregular triangle is the scalene triangle. Scalene triangles are abnormal in that they are distinct by what they are not. The majority triangles drained at chance would be scalene. The angles of the interior of the scalene triangle are normally all different.

Irregular triangle:

In the triangle all the sides are dissimilar length. The sides are not equal and angles are not equal.   The triangle is the general shape of geometry. This is containing the line segment. In the Euclidean geometry three non collinear positions is the represented to the unique triangle and unique or the unique plane. The right triangle is the other name for the right angled triangle. Hypotenuse triangle is the larger size of the right triangle. The length of the square is the equal to the length of the square of the hypotenuse.
Area of the irregular triangle:
The area of a triangle is usually computed as
               area = bh /2
Where b means base length and h means base height.
Calculating the length the side using the formula:
                 S= (a +b +c)/2

Irregular triangle fact:
Shortest side is opposite to the small angle:

• The straight side is constantly differing the minimum interior angle.
• The greatest side is constantly opposite the maximum interior angle.

Longest side is opposite to the largest angle:

• The straight side is constantly differing the nominal interior angle.
• The top side is forever differing the main interior angle.

Example problem:

Example 1:

Triangle EFG shown below is inscribed inside a square of side 40 cm. Find the area of the triangle.

Solution:
Area of triangle = (1/2) base *
= (1/2)(40)(40)
= 800 cm ²
Problem 
Calculating the third side from the specified right scalene triangle

Solution:
GI²=GH²+HI²
150=6²+HI²
150-36=HI²
114=HI²
HI² = `sqrt(114)`
HI = 10.67

Problem 3:
Fine the area of scalene triangle the problem, the given data :  s=6, a=5, b=3, c=4
Solution:
Area formula = `sqrt(s(s-a)(s-b)(s-c))`
 = sqrt(6(6-5)(6-3)(6-4))`
=sqrt(6(1)(3)(2))`
=sqrt(36)`
= 6
The area is 6cm.

Pitch Diameter Formula

When cutting screw threads the mechanic  must have some way to check the threads. Threads should be measured at the pitch diameter. The pitch diameter is a point along the flank or angular surface of the thread

. The thread pitch micrometer is an excellent and easy way to measure threads, but one pitch micrometer will not measure all of the different thread types and sizes. The thread ring gage will tell you whether the thread is right or wrong, but it will not give you an accurate account of the thread depth. The three-wire method is considered to be one of the most accurate and versatile ways of measuring threads. The three-wire method uses three lapped and polished wires and a micrometer to measure the pitch diameter of the thread

The wires are placed in the threads and a micrometer is used to measure over the top of the wires. Different thread types and thread sizes require different size wires. The three-wire method of thread measurement can be used for all types of threads including Unified, Acme, and Buttress thread forms.

To help you understand the three-wire method of thread measurement, we will use an example thread of 1.250-5 ACME-2G . The first piece of information we need to acquire is the "best wire size." The best wire size is the size of the wire that will contact the thread at the pitch diameter. The best wire size can be calculated using a formula, Begin by looking in the index under Thread, Acme, wire method of testing. Turn to this section and find the table which gives the wire sizes for measuring Acme threads with lead angles less than 5 degrees. Now look for the column that shows 5 threads per inch. You should find that the table gives you 0.10329 as the best wire size.  The table also gives you minimum and maximum wire size limits. Make a note of the best wire size.

Now we need to find out what the measurement over the wires should be. Within the same unit you will find a paragraph entitled Three Wire Measurement of Acme and Stub Acme Thread Pitch Diameters. Within this paragraph it will tell you where to find the Approximate Three Wire Formula. The Approximate Three Wire Formula "SIMPLIFIED VERSION" should read as follows:

M = E - 0.86603P + 3W

Where:

     M = the measurement over the wires

     E = the pitch diameter

     P = the pitch or (1 divided by the number of threads per inch)

    W= the wire size

We really only need to find the pitch diameter to solve the formula. We have the wire size. We know the pitch is 1/5 or 0.2 . To find the pitch diameter, go back to the index in the Machinery’s Handbook. Look in the index for Screw threads and threads systems. Under Screw threads and threads systems find Acme. Turn to this section and find the table that gives the Limiting Dimension for American National Standards for General Purpose Acme Single-Start Screw threads. In this table find the Class 2G, Pitch Diameter, Maximum and Minimum limiting diameters for the 1.250-5 ACME-2G thread. You should find that the table gives you 1.1210 as the minimum size and 1.1410 as the maximum size for the pitch diameter calculation. We will use the median dimension of 1.1310 for our formula calculation. Now that we have all of the information, let's plug the numbers into the formula.

M = E - 0.86603P + 3W

M = 1.131-(0.86603 X 0.2) + (3 X 0.10329)

M = 1.131 - 0.173206 + 0.30987

M = 1.268 (micrometer measurement over the wires)

Isosceles Triangle Proof

In an isosceles triangle, two sides are equal in length. An isosceles triangle also has two angles of the same measure; namely, the angles opposite to the two sides of the same length; this fact is the content of the Isosceles triangle theorem. Some mathematicians define isosceles triangles to have only two equal sides, whereas others define that an isosceles triangle is one with at least two equal sides. The latter definition would make all equilateral triangles isosceles triangles. (Source : WIKIPEDIA)

Things need to remember for Proofs of isosceles triangle:

For proving isosceles triangle we need to know the following information about  isosceles triangle,
There are two types of isosceles triangles,
1.Normal isosceles triangle
2.Right isosceles triangle.

Properties:
Properties of parts of  isosceles triangle:
1.The two sides of the isosceles tringles are equal.
2.Two base angles has same measure.
3. Angle ratio of the right isosceless triangle is   45:90:45.
4.The side ratio of the isosceles triangle is  1:1:`sqrt(2)`

Problems on isosceles triangle proof:

Problem 1:
Prove that the following triangle is isosceles triangle.

Proof:
Given , The angle We know that the sum of the angles are 180.
So 100 + x +x = 180
      100 + 2x = 180
Subtract 100 0n both sides.
    2x =180 -100
    2x = 80
Divide by x on both sides,
 x = 40
The base anles are equal two 40 .
According to the properties of isosceles triangle,we can determine that the given triangle is isosceles triangle.
Hence the proof.

Problem 2:
Prove that the triangle with the sides 5 : 5 : 5`sqrt(2)` is an isosceles right triangle triangle.
Proof:
Given,The sides of the triangle is 5 , 5 ,5`sqrt(2)`
We know that in a right Angle triangle,the hypotenuse is greater then the legs and it satisfies the pythagorean theorem,
5`sqrt(2)` > 5 , 5
5²+5² = (5`sqrt(2)` )²
25+25 = 25 *2
50 = 50
So the given sides satisfies the pythagoren theorem.So we can say that it is a right triangle.
The given sides are in the ratio of 1:1:`sqrt(2)`
so the given triangle is isosceles triangle.
Hence the proof.

Pyramid with a Trapezoid Base

A pyramid is a building where the outer surfaces are triangular and converge at a point. The base of a pyramid can be trilateral, quadrilateral, or any polygon shape, meaning that a pyramid has at least three outer surfaces (at least four faces including the base). The square pyramid, with square base and four triangular outer surfaces, is a common version. (Source: Wikipedia)

I like to share this Trapezoid Shape with you all through my article.

Pyramid with trapezoid base:
      A pyramid with a trapezoid base is called as trapezoidal pyramid.
     Volume of trapezoidal pyramid = (1 / 3) * ((1 / 2) * (base₁ + base₂) * height

Example problems for pyramid with a trapezoid base

Trapezoid pyramid problem 1:
        A base length of the trapezoid pyramid is 12 cm, 14 cm and its height of the pyramid is 6 cm. Find the volume of the trapezoidal pyramid.
Solution:
     Given base lengths are b₁ = 12 cm, b₂ = 14 cm, and h = 6 cm.
Formula:
   Volume of trapezoidal pyramid = (1 / 3) * ((1 / 2) * (base₁ + base₂) * height
Substitute the given values in the above formula, we get
= (1 / 3) * ((1 / 2) * (12 + 14) * 6 cm³
= (1 / 3) * 78 cm³
= 26 cm³

Answer:
 Volume of the trapezoidal pyramid is 26 cm³

Trapezoid pyramid problem 2:
        A base length of the trapezoid pyramid is 15 cm, 24 cm and its height of the pyramid is 9 cm. Find the volume of the trapezoidal pyramid.
Solution:
     Given base lengths are b₁ = 15 cm, b₂ = 24 cm, and h = 9 cm.
Formula:
Volume of trapezoidal pyramid = (1 / 3) * ((1 / 2) * (base₁ + base₂) * height
 Substitute the given values in the above formula, we get
= (1 / 3) * ((1 / 2) * (15 + 24) * 9 cm³
 = (1 / 3) * 175.5 cm³
                                               = 58.5 cm³
Answer:
 Volume of the trapezoidal pyramid is 58.5 cm³

Trapezoid pyramid problem 3:
        A base length of the trapezoid pyramid is 6 cm, 11 cm and its height of the pyramid is 5 cm. Find the area of the trapezoid.
Solution:
     Given base lengths are b₁ = 6 cm, b₂ = 11 cm, and h = 5 cm.
Formula:
Area of trapezoid = ((1 / 2) * (base₁ + base₂) * height
Substitute the given values in the above formula, we get
                                               = (1 / 2) * (6 + 11) * 5 cm³
                                               = (1 / 2) * 85 cm³
                                               = 42.5 cm²
Answer:
 Volume of the trapezoidal pyramid is 42.5 cm²

Practice problems for pyramid with a trapezoid base

Trapezoid pyramid problem 1:
        A base length of the trapezoid pyramid is 7 cm, 9 cm and its height of the pyramid is 12 cm. Find the volume of the trapezoidal pyramid.
Answer:
 Volume of the trapezoid pyramid is 32 cm³
Trapezoid pyramid problem 2:
        A base length of the trapezoid pyramid is 10 cm, 22 cm and its height of the pyramid is 7 cm. Find the volume of the trapezoidal pyramid.
Answer:

 Volume of the trapezoid pyramid is 37.33 cm³

Inverting Matrices

The inverse of a square matrix A, sometimes called a reciprocal matrix, is a matrix   such that         

Where I is the identity matrix.

A square matrix A has an inverse if the determinant . A matrix possessing an inverse is called nonsingular, or invertible.

The matrix inverse of a square matrix m may be taken in mathematics using the function inverse[m].

Inverse of 2x2 Matrix

For a 2X2  matrix

A= `[[a,b],[c,d]]`

The matrix inverse is

                  

             =  

                   =    

Inverse of 3x3 Matrix

For a 3X3 matrix

          

The matrix inverse is

A general matrix can be inverted using methods such as the gauss Jordan elimination, Gaussian elimination.

Prism Solve for h

Prism is one of the three dimensional shape. Normally the prisms faces are flat surface. Here we have two bases, one is at the top and another is at the bottom. In prisms we have two kinds of prisms; they are rectangular prism and triangular prism. When a prism bottom is rectangular shape then it’s denoted as rectangular prism. Cube and cuboids are some examples are rectangular prism. When a prism bottom is triangular shape then its said to be triangular prism. 

The diagram given below the structure of prism:

Please express your views of this topic Prism Definition by commenting on blog

Description to prism solve for h:

The following types are used in the prism shape they are

Volume of the regular prism is = BH cubic units,

Where B represents the base area of shape, and H represents height.

The Lateral surface area of the regular prism is given by,

              L.S.A = PH square units.

Where P represents the perimeter of shape and H represents the height of prism

Prism based on rectangular:

The area of the base is calculated by l*w

The perimeter of the base can be calculated by 2(l+w)

Therefore the total surface area is calculated by 2lw+2(l+w)H

Volume of the rectangular prism is V= L * W * H

Triangular based prism:

The area is measured as 1/2 bh

The perimeter of the base is measured by a+b+c.

So the surface are is measured by  bh+ (a+b+c) H

Circular based prism:

The area of the base is calculated as `pi` r²

The perimeter of the base is calculated by 2`pi` r

Therefore the surface area of the prism is 2`pi` r²+2`pi` rH

Let we see some basic problems regarding prisms.

Example 1:

 The dimensions of the rectangular prism is given below ,they are length is 10cm, height is 4 cm, and width is 14 cm.

Solution:

Given dimensions length = 10 cm, height = 4 cm, width = 14 cm.

Formula:

  Surface area = 2lw+2(l+w)H

apply  the values in the above formula, we get

                       = 2 *10*14+2(10+14)4

                       = 240 + 2(24)4

                      = 240 + 48*4

                     = 240 +192

                     = 432 cm²

      Volume = l * w * h  

                      = 10 * 4 * 14 cm³

                      = 560 cm³

Answer:

Surface area = 432 cm²

Volume = 560 cm³

Examples problems to solve h in prism

Example 1:

solve the h value whose prism volume is 1210 cubic units and base is 21 cm.

Solution:

The volume of the prism is V = 1210cubic units.

Base of the prism is = 21cm

                                   V= BH

apply the values in the formula we get

                              1210 = 121*H

                                     h = 1210 /121

                                     h = 10cm.

Example2:

The volume of the  rectangular prism is 360 cm³ and its length and widths are 10 ,4 cm , solve the h value

Solution:

The volume of the rectangular prism is V = 360cm³

Length of the rectangular prism is l = 10cm

Width of the rectangular prism is W = 4 cm

Apply these values in formula we get

                                     360 = 10*4*h

                                      360 = 40 *h

                                     h = 360 / 40

                                    h = 9 cm

Sample problems:

1) The volume of a regular prism is  120cm³.The base of the regular prism is 10 cm , solve its height h?

solution : height h = 12 cm.

2) The volume of the rectangular prism is 420cm3 and its length = 7 cm and width =6 cm and solve its height h?

solution: height h = 10 cm.

Conic Tutorial

The sections of a cone are  ellipses, Circles, parabolas and hyperbolas. Because these diagrams can be obtained by passing a plain through vertex of the double-napped cone. In this conic tutorial we are going to see the sections of the cone.

Circle - conic tutorial:
The circle in the section of a cone is like ellipse. When two foci are coincided to each other the circle is formed. This conic section can be formed when a circular cone is intersected with a perpendicular plane.

Sections of a cone - Parabolas - conic tutorial:

One of the section of the cone is the parabola. The points will not be on the same line but in the plain are equal from a fixed line and in a fixed point.

Solve Maxima and Minima:
The local minimum and the local maximum value must be known for finding the maxima and minima values.

Ellipse - conic tutorial:
The section of a cone is an ellipse. It is from a constant two fixed point with the sum of its focus of all the points. The concept of an ellipse makes different terminologies like focus of ellipse, loci of ellipse, etc.

Equation of ellipse:
Equation of ellipse is `(x - h)^2 / a^2 ` + `(y - k)^2 / b^2` = 1 . here real numbers are h, k, a and b.

Sections of a cone-General Equation of an Ellipse

The points of locus will be F ( p, q ) and the equation of directrix ax + by + c = 0. This is the general equation of ellipse

Theorem on Line contact with an Ellipse:
The condition for the line y = m x + c to touch the ellipse `(x^2/a^2)+(y^2/b^2) ` = 1 is that c = ±  `sqrt(a^2 . m^2 + b^2)` .

Finding the point of contact for Ellipse: 
The tangent y = m x + `sqrt(a^2 . m^2 + h^2)`  touches the ellipse `(x^2/a^2)+(y^2/b^2) `= 1, at the point `[(-a^2m/sqrt(a^2m^2+b^2) , b^2/sqrt(a^2m^2+b^2)]`

Hyperbola - conic tutorial:

A hyperbola is the section of a cone. A conic which an eccentricity greater than 1. It has two branches extending up to infinity, two foci, two directions, two vertices and two axes. ( i . e . )  transverse and conjugate and has a center which is the midpoint of the line of two foci and perpendicular conjugate axis is the line through the center and to the transverse axis.

Two Non Intersecting Lines

The two lines are concurrent means then the pencil is formed by the set of lines passing in their common point: if they are parallel, by the set of parallels to both, in the same direction means, they are non-intersecting lines. In fact, if the lines meet in the ordinary sense of the term or are parallel, the point can be at once obtained. These are called non-intersecting lines.

Example problem for two non intersecting lines:

1. Find the equation for two non intersecting lines that passes through the two points:
     Y = x – 5 and y = x + 5.
     Two non intersecting lines graph for the following equation:
     Y = x - 5
Solution:
   Now plug x values for 0, 1, 2, 3, 4
   When x = 0                                                 
   Y = (0) -5
   Y = 0-5
   Y = -5

   The co ordinate’s point is (0,-5).
   When we plug x = 1
   We get
   Y = 1 -5
   Y = -4

   The co ordinate’s point is (1, -4).
   When we plug x = 2
   We get
   Y = 2 -5
   Y = 2-5
   Y = -3

   The co ordinate’s point is (2,-3).
    When we plug x = 3
    We get
    Y = 3 -5
    Y = -2

   The co ordinate’s point is (3, -2).
    When we plug x = 4
    We get
    Y = 4-5
    Y = -1
The co ordinate’s point is (4, -1).

X	0	1	2	3	4
Y	-5	-4	-3	-2	-1

Two non intersecting lines graph for the following another equation:
   Y = x + 5
Solution:
    Now plug x values for 0, 1, 2, 3, 4
    When x = 0                                                 
    Y = 0+5
    Y = 0+5
    Y = 5

    The co ordinate’s point is (0, 5).
    When we plug x = 1
    We get
     Y = 1 +5
     Y = 6

    The co ordinate’s point is (1, 6).
     When we plug x = 2
     We get
     Y = 2 +5
     Y = 7

     The co ordinate’s point is (2,7).
     When we plug x = 3
     We get
     Y = 3 +5
     Y = 8

     The co ordinate’s point is (3, 8).
     When we plug x = 4
     We get
     Y = 4+5
     Y = 9

     The co ordinate’s point is (4, 9).

X	0	1	2	3	4
Y	5	6	7	8	9

The non intersecting lines represent in graph is,

Practice problem for two non intersecting lines:

1. Find the equation for two non intersecting lines that passes through the two points:
    Y = x – 4 and y = x + 4.
2. Find the equation for two non intersecting lines that passes through the two points:
    Y = x + 8 and y = x - 8.
3. Find the equation for two non intersecting lines that passes through the two points:

    Y = x + 2 and y = x - 2.

Trinomial Notes

Trinomial notes is a polynomial with three monomial terms. In trinomial notes, sum of three monomials is said to be  trinomial. For example, 2x5 – 3x2 + 3 is a trinomial . An Algebraic expression of the form axn is called a monomial in x, The sum of two monomials is called a binomial and the sum of three monomials is called a trinomial.  sum of the finite number of a monomials in term x is called a polynomial of x.

Please express your views of this topic Trinomial Solver by commenting on blog.

How To Factor trinomial Notes:

Example 1:
Factor   trinomial notes x² + 3x − 10.

Solution.
The binomial factors will have this form:
(x   a)(x   b)
What are the factors of 10?  Let us take that they are 2 and 5:

x² + 3x − 10 = (x   2)(x   5).

We have to know how to choose the signs so that,  coefficient for middle term  will be sum of the outers plus the inners -- will be +3.                                  Choose factors , +5 and −2.

=  x² + 3x − 10

= (x − 2)(x + 5)
When 1 is the coefficient for x², the order of  factors it does not matter.

(x − 2)(x + 5) = (x + 5) (x − 2) are same factors.

Example Problems In trinomial notes:

Example 1:
Find factors for trinomial notes x² − x − 12.

Solution:
We must finding factors for 12 whose algebraic sum will be  coefficient of x is −1.
Choose −4 and + 3:
x² − x − 12 = (x − 4 )(x + 3). are facors

Example 2:

Find factors for trinomial notes    x2 + 5xy + 4y2
Solution:
x2+5xy+4y2

=x2+xy+4xy+4y2

=x(x+y)+4y(x+y)

=(x+y)(x+4y)

Example 3:

Find factors for trinomial notes    2a2 - 7ab + 6b2
Solution:
2a2-7ab+6b2
=2a2-3ab-4ab+6b2

=a(2a-3b)-2b(2a+3b)

=(2a-3b)(a-2b)

Example 4:

Find factors for trinomial notes 4x2 - 13xy + 3y2
Solution:
4x2-13xy+3y2
=4x2-xy-12xy+3y2

=x(4x-y)-3y(4x-y)

=(4x-y)(x-3y)

Acute Angles Pictures

Here we are going to see about the introduction to angles. The angle is referred as a figure which is formed by distribution of  two rays with a common point. This common point is referred as end point. The word angle is from the Latin word angulus which defines the point in corner. There are many types of angles in this we are going to see about the acute angles.

The acute angle is the type of angle which measures the angle between 0 to 90 degree and less than the 90 degree. The normal picture of the acute angle is given as,

Pictures of acute angle:
            The acute angles are referred by using the pictures which measures the angle less then 90 degree. The pictures of the acute angles are given below as examples.
Picture 1:

This is the picture of acute angle which defines that the angle measured in this picture is 48⁰ which is less than 90 degree.

Picture 2:

This is other picture of acute angle which defines that the angle measured in this picture is 28⁰ which is less than 90 degree.

Picture 3:

This is the picture of acute angle which defines that the angle measured in this picture is 37⁰ which is less than 90 degree.

Picture 4:

This is also a picture of acute angle which defines that the angle measured in this picture is 63⁰ which is less than 90 degree.

Picture 5:

This is also a picture of acute angle which defines that the angle measured in this picture is 49⁰ which is less than 90 degree.

Picture 6:

This is also a picture of acute angle which defines that the angle measured in this picture is 72⁰ which is less than 90 degree.

Example problems

Problem 1:
When the angle a-180⁰ is an acute angle. what is the value of a?
Options:
a) 240⁰
b) 300⁰
c) 290⁰
d) 320⁰

Solution:
Acute angle is the angle which measures less than 90 degree or 90 degree.
When 240⁰-180⁰ = 60⁰.
Hence the value of a is 60⁰.

Problem 2:
When the angle y - 90⁰ is an acute angle. what is the maximum value of y?
Options:
a) 170⁰
b) 200⁰
c) 100⁰
d) 540⁰

Solution:
Since the acute angle measures less than 90 degree. The maximum value of y is 170⁰.
Where 200⁰- 90⁰ = 110⁰, since it is greater than 90 degree this is not an acute angle.  
100⁰- 90⁰ = 10⁰, since it is less than 90 degree this is an acute angle but it is the minimum value.

540⁰- 90⁰ = 450⁰, since it is greater than 90 degree this is not an acute angle.  
170⁰- 90⁰ = 80⁰, since it is less than 90 degree this is an acute angle.    

Finding Volume in Math

Finding volume in math article deals with the definition of volume and the formula for the various shapes and the model problems related to volume of various shapes.

Looking out for more help on Units of Volume in algebra by visiting listed websites.

Definition of Volume:

Volume is always measured in cube.units. It is defined as the space that is occupied by the entire three-dimensional shape.

Volume formula for various shapes in math.

Volume of the cube is a3 cubic units

Here “a” is the side of the cube.

Volume of the rectangular prism is L*B*H cubic units

Here L is length, B is breadth and H is the height.

Volume of the cylinder is BH cubic units

Here B is the base area and H is the height

Volume of the pyramid is `(1/3)` BH cubic units.

Here B is the base area and H is the height

Volume of the cone is `(1/3) ` BH cubic units.

Here B is the base area and H is the height

Model problem for finding volume in math.

Problem:

1.  Finding volume of the cube when the side measures about 5 cm in length.

Solution:

Since it is a cube, the formula

Volume of the cube is a3 cubic units

Here “a” is the side of the cube.

Here a= 5cm

The volume of the cube =53

=125 cm3

The volume of the cube is125cm3

2. Finding volume of the rectangular prism when the length = 8cm, breadth = 5cm and the height is 4cm.

Solution:

Since it is a rectangular prism, the formula

Volume of the rectangular prism is L*B*H cubic units

Here L is length, B is breadth and H is the height
L= 8cm

B= 5cm

H = 4cm

Volume of the rectangular prism = (8* 5*4) cubic units.

= 160 cm3

Volume of the trectangular prism is 160cm3

3. Find the volume of the cylinder whose base area is 46m^2 and the height of the cylinder is 10m

Solution:

Base area of the cylinder is 46m2

Height of the cylinder is 10m

Formula:

Volume of the cylinder is BH cubic units

Here B is the base area and H is the height

= 46*10

= 460 m3

the volume of the cylinder is 460m3

LENGTH OF MEASURING SYSTEM

The unit of length is meter. In length of measuring system length can be measured by using various terms such as millimeter, centimeter, decimeter, meter, decameter, hectometer and kilometer.

Some of the length of measuring system

10 millimeter is equal to 1 centimeter

10 centimeter is equal to 1 decimeter

10 decimeter is equal to 1 meter

10 meter is equal to 1 decameter

10 decameter is equal to 1 hectometer

10 hectometer is equal to 1 kilometer

By knowing these values we can convert the measures of the length.

Similarly some other important conversions of length of measuring system is

1000 millimeter is equal to 1 meter

1000 meter is equal to 1 kilometer

Similarly inches, feet, yards and miles are also the measurements of length of measuring system.

CONVERSION TABLES OF LENGTH OF MEASURING SYSTEM

1 feet is equal to 12 inches

1 yard is equal to 36 inches

1 yard is equal to 3 feet

1 mile is equal to 5280 feet

We have to divide when converting a larger unit and similarly when we have to convert a smaller unit we have to multiply it.

EXAMPLES ON LENGTH OF MEASURING SYSTEM

Example 1: Convert 5000 meters to kilometer

Solution

We know that 1000 meter = 1 kilometer

So

5000 meter = 5000/1000 = 5 kilometer



Example 2: How many centimeters equal to 450 meter?

Solution:

1 meter = 100 centimeter
so

450 meter = 450 * 100

= 450000 centimeter

Example 3: Convert 6 feet to yards and inches

Solution

We know that 1 feet = 12 inches

3 feet = 1 yard

So

6 feet = 6/3 = 2 yards

6 feet = 6 * 12 = 72 inches

Example 4: Convert 5 miles into feet’s

Solution

1 mile = 5280 feet

So 5 miles = 5 * 5280

= 26400 feet’s.

PROBLEMS ON LENGTH OF MEASURING SYSTEM

Convert 5 feet to inches

Convert 20 yards into feet

Convert 100 meter to millimeter

Convert 52 kilometer to decameter

Convert 3 kilometer to millimeter

Answers:

60 inches

60 feets

100000 mm

5200 decameter

3000000 mm

Coplanar Geometry

Three or more points, lines or any other geometric shapes that lie on the common plane are knows as Coplanar.

Geometric substance lying in a same plane are said to be coplanar. In a plane three noncollinear points are some extent coplanar. Four points are coplanar, defined by them is 0, lying in the same plane. Example, any set of three points in plane are coplanar. Let us see coplanar geometry in brief.

Conditions for coplanar:

Coplanarity is corresponding to the statement that the pair of lines determined by the four points is not skew, and can be equivalently stated in vector form as

`|[x1,y1,z1,1],[x2,y2,z3,1],[x3,y3,z3,1],[x4,y4,z4,1]| = 0`

(x3 - x1).[(x2 - x1) × (x4 - x3)] = 0

The coplanar not only for four points it is also for two or three points.

An arbitrary number of n  points x1 , ..., xn can be checked for coplanarity by finding the point-plane distances of the points x4, ...,xn from the plane determined by (x1,x2,x3), and checking if they are all zero. Therefore, the points are all coplanar.

A set of n vectors v is coplanar if the nullity of the linear mapping defined by v  has dimension 1, the matrix rank of v (or equivalently, the number of its singular values) is n-1.

Parallel lines in three-dimensional space are said to be coplanar, but skew lines are not.

In this article we see the coplanar of lines on same plane. Using the line equation we find the coplanar for the three lines on a same plane.

Example problem for coplanar:

Example: prove that three lines are coplanar, equation lines are 3x + 2y = 0, 3x + 3y = 3, 2x + 2y = 2

Solution:

Given: 3x + 2y = 0 -------------(1)

3x + 3y = 3 -------------(2)

2x + 2y = 2 --------------(3)

`|[x1,y1,c],[x2,y2,c],[x3,y3,c]| = 0`

`|[3,2,0],[3,3,3],[2,2,2]| = 0`

3[(3×2) - (3×2)] - 2[(3×2) - (3×2)] +0[(3×2) - (3×2)] = 0

3(6 - 6) -2(6 - 6) -0(6 - 6) =0

0 = 0

Hence proved thus the three lines are coplanar.

To Logarithmic Table

Logarithmic Properties plays an important role in complex calculations in math. We can perform big calculations in math,physics , engineering using logarithms. It give accurate answer as the calculator. These calculations are carried out with the help of logarithmic table.We will see the logarithmic table. The ways of reading the logarithmic table and how to use it in calculations.

Any number x in standard form is written as x = m x 10^p where 1 `<=` m<10 p="">Taking log on both sides we get
log₁₀x = log₁₀(m x 10p) = log₁₀m + plog₁₀10
             = log₁₀m + p
Here p is the characteristic of log x and log₁₀m is called the mantissa of logx

How to find the logarithm of a number:

• Step 1: Write the number in the standard form.

For example 431.5 = 4.315 x 10²

• Step 2: Find the characteristic p of the logarithm.

Here p = 2

• Step 3: Find the mantissa from the table.

To find the log of 4.315 from the table. log 4.31 is 0.634473 `~~` 0.6350. We take the approximate value.

log 4.315 = p + logm = 2+0.6350 = 2.6350

Logarithmic table from 1 to 4.99

1.000	0.00000000			2.00	0.3010300	3.00	0.4771213	4.00	0.6020600
1.001	0.00043408			2.01	0.3031961	3.01	0.4785665	4.01	0.6031444
1.002	0.00086772			2.02	0.3053514	3.02	0.4800069	4.02	0.6042261
1.003	0.00130093			2.03	0.3074960	3.03	0.4814426	4.03	0.6053050
1.004	0.00173371			2.04	0.3096302	3.04	0.4828736	4.04	0.6063814
1.005	0.00216606			2.05	0.3117539	3.05	0.4842998	4.05	0.6074550
1.006	0.00259798			2.06	0.3138672	3.06	0.4857214	4.06	0.6085260
1.007	0.00302947			2.07	0.3159703	3.07	0.4871384	4.07	0.6095944
1.008	0.00346053			2.08	0.3180633	3.08	0.4885507	4.08	0.6106602
1.009	0.00389117			2.09	0.3201463	3.09	0.4899585	4.09	0.6117233
1.010	0.00432137	1.10	0.0413927	2.10	0.3222193	3.10	0.4913617	4.10	0.6127839
1.011	0.00475116	1.11	0.0453230	2.11	0.3242825	3.11	0.4927604	4.11	0.6138418
1.012	0.00518051	1.12	0.0492180	2.12	0.3263359	3.12	0.4941546	4.12	0.6148972
1.013	0.00560945	1.13	0.0530784	2.13	0.3283796	3.13	0.4955443	4.13	0.6159501
1.014	0.00603795	1.14	0.0569049	2.14	0.3304138	3.14	0.4969296	4.14	0.6170003
1.015	0.00646604	1.15	0.0606978	2.15	0.3324385	3.15	0.4983106	4.15	0.6180481
1.016	0.00689371	1.16	0.0644580	2.16	0.3344538	3.16	0.4996871	4.16	0.6190933
1.017	0.00732095	1.17	0.0681859	2.17	0.3364597	3.17	0.5010593	4.17	0.6201361
1.018	0.00774778	1.18	0.0718820	2.18	0.3384565	3.18	0.5024271	4.18	0.6211763
1.019	0.00817418	1.19	0.0755470	2.19	0.3404441	3.19	0.5037907	4.19	0.6222140
1.020	0.00860017	1.20	0.0791812	2.20	0.3424227	3.20	0.5051500	4.20	0.6232493
1.021	0.00902574	1.21	0.0827854	2.21	0.3443923	3.21	0.5065050	4.21	0.6242821
1.022	0.00945090	1.22	0.0863598	2.22	0.3463530	3.22	0.5078559	4.22	0.6253125
1.023	0.00987563	1.23	0.0899051	2.23	0.3483049	3.23	0.5092025	4.23	0.6263404
1.024	0.01029996	1.24	0.0934217	2.24	0.3502480	3.24	0.5105450	4.24	0.6273659
1.025	0.01072387	1.25	0.0969100	2.25	0.3521825	3.25	0.5118834	4.25	0.6283889
1.026	0.01114736	1.26	0.1003705	2.26	0.3541084	3.26	0.5132176	4.26	0.6294096
1.027	0.01157044	1.27	0.1038037	2.27	0.3560259	3.27	0.5145478	4.27	0.6304279
1.028	0.01199311	1.28	0.1072100	2.28	0.3579348	3.28	0.5158738	4.28	0.6314438
1.029	0.01241537	1.29	0.1105897	2.29	0.3598355	3.29	0.5171959	4.29	0.6324573
1.030	0.01283722	1.30	0.1139434	2.30	0.3617278	3.30	0.5185139	4.30	0.6334685
1.031	0.01325867	1.31	0.1172713	2.31	0.3636120	3.31	0.5198280	4.31	0.6344773
1.032	0.01367970	1.32	0.1205739	2.32	0.3654880	3.32	0.5211381	4.32	0.6354837
1.033	0.01410032	1.33	0.1238516	2.33	0.3673559	3.33	0.5224442	4.33	0.6364879
1.034	0.01452054	1.34	0.1271048	2.34	0.3692159	3.34	0.5237465	4.34	0.6374897
1.035	0.01494035	1.35	0.1303338	2.35	0.3710679	3.35	0.5250448	4.35	0.6384893
1.036	0.01535976	1.36	0.1335389	2.36	0.3729120	3.36	0.5263393	4.36	0.6394865
1.037	0.01577876	1.37	0.1367206	2.37	0.3747483	3.37	0.5276299	4.37	0.6404814
1.038	0.01619735	1.38	0.1398791	2.38	0.3765770	3.38	0.5289167	4.38	0.6414741
1.039	0.01661555	1.39	0.1430148	2.39	0.3783979	3.39	0.5301997	4.39	0.6424645
1.040	0.01703334	1.40	0.1461280	2.40	0.3802112	3.40	0.5314789	4.40	0.6434527
1.041	0.01745073	1.41	0.1492191	2.41	0.3820170	3.41	0.5327544	4.41	0.6444386
1.042	0.01786772	1.42	0.1522883	2.42	0.3838154	3.42	0.5340261	4.42	0.6454223
1.043	0.01828431	1.43	0.1553360	2.43	0.3856063	3.43	0.5352941	4.43	0.6464037
1.044	0.01870050	1.44	0.1583625	2.44	0.3873898	3.44	0.5365584	4.44	0.6473830
1.045	0.01911629	1.45	0.1613680	2.45	0.3891661	3.45	0.5378191	4.45	0.6483600
1.046	0.01953168	1.46	0.1643529	2.46	0.3909351	3.46	0.5390761	4.46	0.6493349
1.047	0.01994668	1.47	0.1673173	2.47	0.3926970	3.47	0.5403295	4.47	0.6503075
1.048	0.02036128	1.48	0.1702617	2.48	0.3944517	3.48	0.5415792	4.48	0.6512780
1.049	0.02077549	1.49	0.1731863	2.49	0.3961993	3.49	0.5428254	4.49	0.6522463
1.050	0.02118930	1.50	0.1760913	2.50	0.3979400	3.50	0.5440680	4.50	0.6532125
1.051	0.02160272	1.51	0.1789769	2.51	0.3996737	3.51	0.5453071	4.51	0.6541765
1.052	0.02201574	1.52	0.1818436	2.52	0.4014005	3.52	0.5465427	4.52	0.6551384
1.053	0.02242837	1.53	0.1846914	2.53	0.4031205	3.53	0.5477747	4.53	0.6560982
1.054	0.02284061	1.54	0.1875207	2.54	0.4048337	3.54	0.5490033	4.54	0.6570559
1.055	0.02325246	1.55	0.1903317	2.55	0.4065402	3.55	0.5502284	4.55	0.6580114
1.056	0.02366392	1.56	0.1931246	2.56	0.4082400	3.56	0.5514500	4.56	0.6589648
1.057	0.02407499	1.57	0.1958997	2.57	0.4099331	3.57	0.5526682	4.57	0.6599162
1.058	0.02448567	1.58	0.1986571	2.58	0.4116197	3.58	0.5538830	4.58	0.6608655
1.059	0.02489596	1.59	0.2013971	2.59	0.4132998	3.59	0.5550944	4.59	0.6618127
1.060	0.02530587	1.60	0.2041200	2.60	0.4149733	3.60	0.5563025	4.60	0.6627578
1.061	0.02571538	1.61	0.2068259	2.61	0.4166405	3.61	0.5575072	4.61	0.6637009
1.062	0.02612452	1.62	0.2095150	2.62	0.4183013	3.62	0.5587086	4.62	0.6646420
1.063	0.02653326	1.63	0.2121876	2.63	0.4199557	3.63	0.5599066	4.63	0.6655810
1.064	0.02694163	1.64	0.2148438	2.64	0.4216039	3.64	0.5611014	4.64	0.6665180
1.065	0.02734961	1.65	0.2174839	2.65	0.4232459	3.65	0.5622929	4.65	0.6674530
1.066	0.02775720	1.66	0.2201081	2.66	0.4248816	3.66	0.5634811	4.66	0.6683859
1.067	0.02816442	1.67	0.2227165	2.67	0.4265113	3.67	0.5646661	4.67	0.6693169
1.068	0.02857125	1.68	0.2253093	2.68	0.4281348	3.68	0.5658478	4.68	0.6702459
1.069	0.02897771	1.69	0.2278867	2.69	0.

Algebra is widely used in day to day activities watch out for my forthcoming posts on Variance Statistics Formula and Definition of least Common Multiple. I am sure they will be helpful.