Descartes Cartesian Coordinate System

Introduction for descartes Cartesian coordinate system:

In Cartesian coordinates system,name Cartesian is derived from the name of french mathematician and philosopher René Descartes (Latin: Cartesius) from the Euclidean geometry.Cartesian coordinate system which consist of analytic geometry, calculus, and cartography. Descartes introduces the new  point  or plane  on a surface by two intersecting axes.In analytical geometry solving we have point coordinates with the formulas for finding the required parameters and in this section we have the points and the problems.

Descartes Cartesian Coordinates System:

In descartes Cartesian coordinates system we have the point geometry and the formulas for finding the parameters.In a Cartesian coordinates we have many points such as collinear point coordinates, equidistant point coordinates and mid point coordinates.

Collinear points:
Collinear points are a point when three or more points lies on same line.

Midpoint:
Mid point is a halfway point where the line segment divided into two equal parts.

Equidistant point:
In a line segment a point is equal length from other points which are in congruent then the point are equidistant point.I like to share this second order differential equation with you all through my article.

Problems in Descartes Cartesian Coordinates System:

Example 1:
Find the Cartesian coordinates distance between the points A(2,6) and B (2,3).

Solution:
Let assume "d" be the distance between A and B.           (x1,y1)= (2,6), (x2,y2)= (2,3)

Then d (A, B) = `sqrt((x_2-x_1)^2+(y_2-y_1)^2)`

= `sqrt((2-2)^2 +(3-6)^2)`

= `sqrt(0^2+(-3)^2)`

= `sqrt(0+9)`

=`sqrt9`

=3
Example 2:
Determine the mid point coordinates  of a line segment joining given points A(1,8) and B(-1,-5)

Solution:
The required mid point is
Formula = ` ((x_1+x_2)/(2))`,`((y_1+ y_2)/(2))` here, (x1, y1) = (1,8),(x2, y2) = (-1,-4)

= `((1-1)/(2))``((8-4)/(2)) `

= ` (0/2)`` (4/2)`

= (0,2)

Example 3:
Find the centroid coordinates of a triangle whose vertices's points are given (0, -3), (0,6) and (3,3).

Solution:
(x1 y1) = (0,-3),  (x2 y2) = (0,6), (x3 y3) =(3,3)

Formula for centroid =   ` ((x_1+ x_2+ x_3)/(3))` ,`(( y_1+ y_2+ y_3)/(3))`

The centroid of the triangle  = `((0+0+3)/3)`,`((-3+6+3)/3)`

=`(3/3),(6/3)`

=(1,2)

Computing Rational Expressions

Definition of computing rational expressions:
A rational expression is simplified, or reduced to lowest terms, if its numerator and denominator have no common factors other than 1 and -1. If a rational expression does contain common factors, we use the properties of the real number system to write

`ac/bc`=`a/b``xx``c/c`=`a/b``xx`1=`a/b`   (a, b, c are real number, and bc`!=`0.)

This process is often called “canceling common factors.” To indicate this process, we often write

`a/b`=`a/b`

Rules for Computing Rational Expressions:
Rule 1: Steps for computing rational expressions

Step 1: Set the terms containing the identical variable collectively in algebraic exponential expressions.

Step 2: Accomplish the operation inside the parentheses for the variable and other.

Step 3: Revise the rational expressions and to simplifying rational expressions.

Step 4: To make sure rational expressions, if there is able to simplify rational expressions and then repeat the step 1 to 4.

Rule 2: Order of operation for computing rational expressions

In long math problems with +,-,x,%,(), and exponents in them, you have to identify what to do first. Without follow the similar rules, you may get unlike answers. You can easily keep in mind the silly sentence, Big Elephants Destroy Mice And Snails, you can commit to memory the order of operations, and you must follow.

Big                     “B” means Brackets. We need to carry out operation in side parentheses first.

Elephants         “E” means an exponent; you must calculate exponents next in expressions by j.

Destroy             “D” means division

Mice                  “M” means multiply begin on the left of the equation and perform all divisions and multiplication in the order in which they appear.

And                    “A” means addition

Snails               “S” means subtract. For all time on the left hand side and accomplish additions and subtractions operation.

Rule 3: study for computing rational expressions with exponents.

The rules are given by

`x^m` `xx` `x^n` = `x^(m+n)`

`((x^(m))/(x^(n)))` =  `x^(m-n)`

`((x^(m))^n)` = `x^(mn)`

`(x y)^m` = `x^m` `y^m`

`(x/y)^n` = `(x^(n))/(y^(n))`

`x^(-n)` = `1/(x^(n))`

`(x/y)^(-n)` = `(y/x)^(n)`

Where quantities in the denominator are taken to be nonzero in computing rational expressions, Special cases include

`x^1`=`x`

And

`x^0=1`

For  x`!=`0. The definition `0^o`=1 is sometimes used to simplify formulas, but it should be kept in mind that this equality is a definition and not a fundamental mathematical truth.

Example Problem for Computing Rational Expressions:

To solving computing rational expressions using above rules, ( 8`V^(2)` `-:``2V)+5`Y^(2)``xx``Y^(4)`

Solution:

Step 1: Is to group like terms. Set the terms containing the same variable jointly. Set constants collectively and brackets.

( 8`V^(2)` `-:``2V)+5`Y^(2)``xx``Y^(4)`

Step 2: Is to accomplish the operation inside the parentheses for the variable V.

( 8`V^(2)` `-:``2V)=(8`-:`2)`xx``(V^(2)``-:`V^(1))`=(`8/2`)`xx``(V^(2-1))`=4`xx``V^(1)`=4V

Step 3: Is to accomplish the operation inside the parentheses for the variable Y using expression with exponents rule.

5`Y^(2)``xx``Y^(4)`=5`xx``(Y^(2+4))`=5`xx``Y^(6)`=5`Y^(6)`

Step 4: Is to revise the given problem.

4V+5`Y^(6)`

Since the left over terms are not like terms, the problem cannot be any further.

The answer is 4V+5`Y^(6)`

Venn Diagram Problem Solution

Introduction to Venn diagram problem solution:

In set theory, the concept of Venn diagram is an essential part of set. The Venn diagram show all theoretically like rational relations among limited groups of sets. These are used for solving the set problems in use of diagrams. Another name of Venn diagram is known as set diagram. Solving the set problems with use of Venn diagram is very easy. We will see few problems on Venn diagram with solutions.

Example Problems:

Problem 1:

In a class of 64 students, 26 students participating in Chess, 35 participating in Tennis, and 8 take both Chess and Tennis. Find how many students in the class are not participating in either Chess or Tennis with use of venn diagram?

Solution:

Given, Total number of students in class = 64

Number of students participating in Chess = 26

Number of students participating in Tennis = 35

Number of students participating in both Chess and Tennis = 8

The venn diagram for these information is,



Here, a = Number of students not participating in either Chess or Tennis.

That is, 18 + 8 + 27 + a = 64

53 + a = 64

a = 64 - 53

a = 11

Therefore, the number of students not participating in either Chess or Tennis is 11 students.Understanding differential calculus is always challenging for me but thanks to all math help websites to help me out.

Problem 2:

There are 70 houses in a city. In that, 48 houses have Washing machine and 32 houses have Refrigerator. Find how many numbers of houses have both Washing machine and Refrigerator with use of venn diagram?

Solution:

Let W = {Number of houses having Washing machine}

R = {Number of houses having Refrigerator}

WnR = {Number of houses having both Washing machine and Refrigerator}

Consider, n(WnR) = a



Therefore, Number of houses having both Washing machine and Refrigerator is,

48 - a + a + 32 - a  = 70

80 - a = 70

-a = 70 - 80

-a = -10

a = 10

Answer: Number of houses having both Washing machine and Refrigerator = 10 houses.

These are all few example problems of using venn diagram with solution.

Algebra Polynomials Function

Introduction of algebra polynomials function:

In the algebra polynomials functions the algebraic expression is look y=an is said to be monomial in y here a is the constant number, y is the variable  and n is the positive integer. The number a is said to be the coefficient of yn and n is the degree of monomial.

Example of polynomial:  7x3 is a monomial in x of degree 3 and 7 is the coefficient of x3. In monomial the sum of finite number in x is called a polynomial in x.

Algebra polynomial function is denoted by the function p(x), here p is the function and x is the variable.

Example of algebra polynomials function:

P(x) = ax^5+bx^4+cx^3+dx^2 +ex + f here we need to find the algebra polynomials function of p(5). X is the variable  and  other alphabet are constant.

Problem on Algebra Polynomials Function:

Problems1:. Find the functions of f(3). f(x) = x^3 +2x^2 + 2x + 4

Solution

Given function f(x)

f(x) = x^3 +2x^2 + 2x + 4 find the f(3).

Here the value of x is given as 3

f(3) = 33 + 2*32 + 2*3 +4

f(3) = 27 +18+ +6 +4 In this step 33 is calculated  as 27 and 3 square is 9

f(3) = 55.

The answer for algebra polynomials functions  f(3) = 55.


Problems Using the Algebra Polynomials Function with High Powers:

Problems 1: Find the fucntions f(5). f(x) = x5+x^4 +x^3 +2x^2 + 2x + 4

Solution

Given the function of  f(x)

f(x) = x5+x^4+x^3 +2x^2 + 2x + 40 find the  function of f(5).

Here the value of x is given as 5

f(5) =55+54+ 53 + 2*52 + 2*5 +40.

f(5)=3125+625+125+50+10 +40 In this step5 power 5 is 3125, 5 power 4 is 625, 5cube is 125 and calculated in the function.

f(5) = 3975.

The answer for algebra polynomials functions f(5) = 3975.

Scientific Method Math

Introduction to scientific method math:
In this article shall we discuss about scientific method math. Here, scientific method explains to the organization of system to examine fact, attain new information, or precise and integrate preceding information. To be appearance scientific, a method of question have to be based on gathering visible, experimental and computable proof subject to precise principles of analysis. Scientific methods include the set of numbers through study and testing, and the formulation and testing of theory.

Steps for Scientific Method Math:

A step for scientific method math is given below that:

The scientific method math is an approach to ask and answer scientific questions through creation comments and doing experiment.

The steps for scientific method math are:

Inquire a Question

Perform Background Research

Build a Hypothesis

Observe Your Hypothesis through Doing an Experiment

Observe Your Data and sketch a Conclusion 

Communicate your outcome

It is significant for your explore to be a fair test. A "fair test" occurs when you modify just one factor (variable) and stay all other situation the same.



Overview for scientific method math:

The scientific method math for testing that is used to find out comments and answer questions. Scientists use the scientific method to seem for reason and result relations in nature. In other words, they plan a research so that changes to one thing cause something else to vary in an expected way.

Now as it does for a specialist scientist, the scientific method will help you to center your science fair scheme question, construct a hypothesis, plan, perform, and estimate your experiment.

Application of Scientific Method Math:

The application of scientific method math is given below that:

How many facts do you know about the sun?

Mass: 3.28 x 1027 tons

Temperature: 2.7 x 106 degrees Fahrenheit

Energy generated per minute: 3.5x 104 horsepower

Application of solving scientific method math:

All the numbers now are writing as products of a number between 1 and 10 and a suitable power of 10. This is also denoted as scientific method. When written in standard method, these numbers are

3,280,000,000,000,000,000,000,000,000

2,700,000

35,000

Regular Polygon Definition

Definition of polygons:
Polygons are the two dimensional closed figure that is made up of three or more than three line segments.

Types of polygons:

regular polygons
irregular polygons
convex polygons
concave polygons
Crossed polygons.

Basic Criteria that Helps in Identifying Polygons:

Identification of regular polygon:

In this polygons

All the sides of the polygons are equal in length

All the angles of the polygons are equal in degrees.

Regular polygons are convex

Identification of irregular polygon

In this polygons

No the sides of the polygons are equal in length

All the angles of the polygons are different in degrees.

Irregular polygons are convex or concave.

Identification of convex polygon:

Here

The internal angle of the polygon is equal to 180 or less than 180 degrees

Identification of concave polygon:

Here

The one or more internal angle of the polygon is greater than 180 degrees

Formula to identify the regular polygons:

Internal angle of regular polygon= `(((M-2)*180)/M)` degrees
External angle of the regular polygon = 180- internal angle of regular polygon.
Number of diagonals in a polygon= `(M (M-3))/2`


Here M is the number of sides of the regular polygon.

Area of regular Polygon = 1/2× M × Radius2 × sin (2 × p/n)
Area of regular Polygon = 1/4 × M× Side2 / tan (p/n)

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Using this Formula, we can Find the Identify the Type of the Polygon.

Model problem 1:

Help to identify the polygon whose sides are equal and the number of the sides is 6 and internal angle is 120 degrees:
Solution:

Number of the sides of the polygon is 6

Internal angle is 120 degrees

All the sides are equal.

From the given condition, we can conclude the given polygon is regular hexagon.

Here the internal angle is less than 180 degrees

So it is a convex one

The polygon is regular hexagon convex polygon.

2.Help to identify the polygon whose sides are equal and the number of the sides is 8 and internal angle is 135 degrees:

Solution:

Number of the sides of the polygon is 8

Internal angle is 135 degrees

All the sides are equal.

From the given condition, we can conclude the given polygon is regular octagon

Here the internal angle is less than 180 degrees

So it is a convex one

The polygon is regular octagon convex polygon.

Solving Volume of a Cylinder

Introduction to solving volume of a cylinder:

Volume is how much three-dimensional space a substance or shape occupies or contains, often quantified numerically using the SI derived unit, the cubic meter. The volume of a container is generally understood to be the capacity of the container . Here we are going to learn about how to solving the volume of a cylinder and its examples.                                                                             (Source from Wikipedia)


Formula for solving the volume of cylinder:



Volume of cylinder = `pi` r 2 h cubic units

Solving Volume of a Cylinder - Example Problems

Example: 1

A cylinder with the radius 14 meter and height is 18 meter. calculate the volume of the cylinder.

Solving steps:

We know that formula for figure out volume of the cylinder is `pi` r2 h

Here the given is h = 18 m, r = 14 m, `pi ` = 3.14

Substitute all the value in the above formula we get

Volume = 3.14 * (14 2) * 18

Simplify the above we get

=3.14 * 196 * 18

= 11077.92

Therefore the volume of tank is 11077.92 m3

Example: 2

A cylinder with the diameter 40 meter and height is 14 meter. Calculate the volume of the cylinder

Solving steps:

Volume of cylinder =  `pi` r2 h

Here the given is diameter so we have to find the radius value

Radius = `("diameter" / 2)`

r = `38/2`

r = 20 m

Now we calculate the volume

r = 20 and h = 14 m substitute the formula we get

Volume = 3.14 * (202) * 14

Simplify the above we get

= 3.14 * 400 * 14

= 17584 m3

Therefore the volume of the cylinder is 17584  m3

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Solving Volume of a Cylinder - Example: 3

A cylinder with the radius 7.9 cm and height is 8.1 cm. calculate the volume of the cylinder

Solving steps:

Formula:

Volume of cylinder = `pi` r2 h

`pi` = 3.14, r = 7.9 cm, h = 8.1 cm substitute this value into the above formula we get

= 3.14 * (7.92) * 8.1

Simplify the above we get

= 3.14 * 62.41 * 8.1

= 1587.33 cm3

Therefore the volume of the tank = 1587.33 cm3