How to calculate Area of Ellipse?

An ellipse is nothing but the planar curve which is resulted by the intersection by a plane of a cone in a way such that it will produce a closed curve. Circles can be said as a special case of ellipse which can be obtained by cutting in orthogonal plane of the cone’s axis. Ellipse can also be said as the locus of all the points in the plane in which the sum of the distances of two fixed points will be a constant. Before going into the formula for area of an ellipse, it is essential to know the elements of ellipse which forms the structure of it. An ellipse is said to be a smooth and closed curve symmetric about its vertical and horizontal axis.

The longest diameter of an ellipse is called as major axis and the shortest diameter of an ellipse is called as the minor axis. These both axes are the lines that pass through the centre of an ellipse. With the measurement of both these axis only, the area of a ellipse will vary. Each of the above said axes bisects perpendicularly with the other. Also, the sum of the distance from the two focus of the ellipse to any point P on the ellipse will be equal to the major axis. It can also be said that each axis cut the other equally into two parts and they cross at right angles to each other. Also, if these both axes are equal in length, then it is called as circle.

Area of an Ellipse Formula
The standard equation for representing the ellipse equation is given as,
X2 / a2 + Y2 /b2 = 1.
This equation is a standard equation in which when the ellipse is centred with origin.

But algebraically the ellipse area formula is given in other terms with the help of semi major axis (half of major axis) and semi minor axis (half of minor axis). Thus the formula for area of ellipse is given as Pi*a*b, where ‘a’ is the semi major axis and ‘b’ is the semi minor axis. This formula is actually arrived from the formula of circle Pi multiplied by radius square. Here the radius is split into semi major and semi minor axis.
Also at some special cases, when an ellipse is given by the implicit equation which is given as,
AX2 + Bxy + CY2 = 1, then the area of the ellipse would be 2*Pi whole divided by square root of 4AC subtracted with B square.

Percentage formula

Percentage is the word which we come across frequently in our daily life. It is mainly used to compare quantities like the percentage of marks obtained in various exam results or to describe interest rates, stock market details, different types of surveys conducted in various areas etc. From this we can say that Percentages are very much a part of everyone’s life irrespective of the field. Let us now understand the meaning of the word Percent, ‘cent’ means hundred and hence it means ‘per hundred’. So the comparison is done ‘per hundred’ and then calculated for the entire group or the total quantities.  In math percent means ‘out of hundred’ or in simple words ‘divide by hundred’ , so if we say 60% it means 60 out of 100 or 60 divided by 100. The symbol or the sign used to show the percentage is ‘%’, this symbol we come across in various discount sales which are displayed in the shops especially in festive seasons, like 30% discount, up to 50% discount on selected items etc.

In the process of figuring percentages there are different methods which can be followed according to the given problem, like they can be converted into fractions or decimal numbers which makes the calculations easy. 25% would be 25/100 in the fraction form and 0.25 in decimal form, so in 25% of 50 we just multiply 0.25 with 50 to get the desired value, 0.25 x 50=12.5

While calculating percentages the Percentage formula is, [part/whole]=[%/100], we need to remember that the number which is after ‘is’ is the ‘part’ and the number that is after ‘of’ is the ‘whole’. For example, Find percentage, __% of 40 is 8. Here whole=40 and part=8; so  8/40= %/100, on simplification we get 25 %. Another formula to calculate percentage is given by, is/of = %/100. Here we cross multiply to get the desired value, which means the numerator of one fraction is multiplied with the denominator of the fraction on the other side of the equal to sign and then the unknown value is calculated, for instance the cross product of a/b=c/d would be ad = bc. Example: 12 % of 50 is? In the given problem %=12, ’is’ = ? and ‘of’=50, which gives ‘is’/50 = 12/100 here we cross multiply and simplify which gives, ‘is’ = 50x12/100 on further simplification we get the final value of ‘is’= 6. So, 12% of 50 is 6.

Operations on Combination of Positive and negative numbers

The numbers have been fascinating to the humankind for so many years. First we started with the natural numbers. The concept of whole numbers emerged after the advent of natural numbers. The number zero was added to the number line with this. Many scientists still don’t agree to the idea of having zero clubbed with whole numbers. Along with whole came the concept of negative numbers. The question was where to accommodate them. This was solved with the introduction of integers.

Then the question how to define integers, with the introduction of negative numbers emerged. This was solved by adding the negative numbers under the umbrella of integers. There can be various integers examples to explain the concept. Positive numbers were called positive integers and negative numbers came to be known as negative integers. Zero also formed the part of integer set. Now let us see how to do integers operations for better understanding. ‘-5’ is an example of negative integer. So, by now one must have understood what is integers and will be able to define it. The need for integers arose when negative numbers came into picture.

The art of teaching integers can be very interesting. The concept of negative numbers being added to the number is very fascinating for the students who were just dealing with positive numbers all the while. Students may need integers help as the concept will be new for beginners. Integers can also be added, subtracted, multiplied and so on as like the other numbers. Both the positive and negative integers can be easily added and subtracted. The only thing is that when we add two negative integers we get a negative integer unlike the addition of two positive integers where we get a positive integer as an answer. The rules for integer operations are very simple and can be explained with the help of examples. When a negative integer is added with a positive integer the sign of the answer depends on the magnitude of the integers. If we add ‘-4’ and ‘2’ the answer would be ‘-2’. It is because the magnitude of negative integer is more. So, we carry out the subtraction operation and the sign of the larger integer is the sign of the answer. If we add ‘-4’       and ‘-2’ we get ‘-6’, i.e. both numbers are added and minus sign put.

Truth Table for dummies

Truth table is a mathematical table used in logic.

Truth table  means which the value of the predicates are true or False. If the values of predicates are always true then we can say that are tautologically implemented.

Ther are mainly four predicates available. They are

(1)AND (Conjuction)

(2)OR (Disjunction)

(3)Implication (Conditional)

(4)If and only if (Biconditional)

These Four type are very important to know the Truth values of the given table.

Prepositions Descriptions of Truth Tables for Dummies

AND (CONJUCTION):

If P and Q are two propositions then 'P and Q' are written as PΛQ is a proposition whose truth table value is TRUE only when both P and Q are TRUE.

OR (Disjunction):

If P and Q are two propostitions then 'P or Q' are written as PνQ is a proposition whose truth table values is FALSE only when both P and Q are FALSE.

IMPLICATION (CONDITIONAL):

If P and Q are any two propositions then 'P implies Q' are written as 'P->Q' is a proposition whose truth table values is FALSE only when P is TRUE and Q is FALSE.

IFF (BICONDITIONAL):

If P and Q are any two propositions then 'P IFF Q' writen as 'P<->Q' is truth table values is TRUE only when both P and Q have same truth value.

Negation:

If P is any Proposotion Then 'Not P' written as' ˜P ' is a proposition whose truth value is FALSE only when P is True.

Example:

If P = 2*3 = 5

Then ˜p = 2*3 ≠ 5

Note:

If the truth table of any proposition is true then we can say that it is Tautologically implemented.

Formulas:

PΛQ = P

PΛQ = Q

These both are simiplication formulas.

Proposition Logic with Truth Tables for Dummies

Proposition:

A declarative sentence to which we can assign.

True or False:

Every Truth table should contains the both true and false values.

Law of excluded Middle:

A proposition can not be simultaneously TRUE and FALSE.

Truth table for AND:

P Q PΛQ
T T T
T F F
F T F
F F F


Truth table for OR

P Q PνQ
T T T
T F T
F T T
F F F

Types of Numbers

There are Different Types of Numbers or Different Kinds of Numbers which we deal in our day today lives. Types of Numbers can be classified into various categories like natural numbers, Integers, rational numbers, irrational numbers, real numbers, even numbers and odd numbers. Natural numbers are just the counting numbers like 1, 2, 3,4,5,6,7,8,9 …etc.

In integers they are similar to natural numbers except we include negative say -2,-1, 0,1,2,3...And we include zero. Basically it is combination of natural numbers and zero and also includes all negative numbers. Let us move on to rational numbers, the number can be written as fraction of a and b, where a and b are integers like the examples as we have studied above, but b is never equal to zero. Let us say 3/2 and this is equal to 1.5 or if we have 4/2 that will result as 2.

Another example is 1/3 which results as 0.3333 repeat infinitely, this is also considered rational. Irrational is defined exactly what the word is that is it’s not rational number, that is irrational.

So it cannot be a fraction, the famous example is pie which is 3.1415… this goes on etc. which never repeats, one more important point in irrational numbers is that they do not repeat and it never terminates or never stops.

Interesting number 22/7 will give us the repeated pattern goes on and on, thus it is called irrational number. Now lets us understand about real numbers which is another Type of Numbers. 

Real numbers are basically any number besides imaginary numbers, any number on the number line can be real, let it be negative one. Let us also learn about even numbers which comes under category of Types of Number, even numbers are those numbers which are divisible by 2 or which can be exactly divided by 2. So let us say if we have number 4, then it is exactly divided by 2, here exactly divided means the remainder is zero.

As 2 times 2 is equals to 4, that is why 4 is an even number. Take 6, 6 also can be easily divided by 2 as 2 times 3 equals 6. Thus all the numbers which can be divided by number 2 or all the multiples of 2 are called even numbers.
All those number which cannot be divided by 2 are the odd numbers. Hence there are different Type of Number.

Frequency table examples

Frequency is the digit of occurrence of a repeat the  event per unit time it is also called temporal frequency. The time period of one cycle in a repeat the event, so the period is the the same of the frequency.
Frequency of waves

Frequency  is  inverse of the concept of wavelength, simply, frequency is inversely proportional to wavelength λ . The frequency f is the same to the phase velocity v of the wave divided by the wavelength λ of the wave:
   f=v/λ, then v=c   f=c/λ

Other Types of Frequency Table Examples:

The Angular frequency ω is define the rate of alteration in the orientation slant, or in the phase of a sinusoidal waveform
Angular frequency is considered in radians per second (rad/s).
The  frequency is analogous to sequential frequency, but the time axis is replace the one or more spatial displacement axis.
Examples on Frequency Table

Consider the frequency distribution shown in scores of 35 students in a mathematics test.

The mean of these scores = total scores /number of boys

 One way would be to add the scores of all the 35 boys as separate addends and then divide the sum by 35.It will be quite cumbersome.we can make the task easier by multiplying each score by its frequence and then dividing by the total number of boys.

frequency table:

Scores(s)      frequency(f)

      6 0                 1

      65                  2

      70                  6

      75                  8

      80                  7

      85                  5

      90                  3

      95                  2

     100                  1

                         35

Then,  x = 60*1+65*2+70*6+75*8+80*7+85*5+90*3+95*2+100*1/35

              = 60+130+420+600+560+425+270+190+100/35

              =  2755 / 35

              = 551 / 7

              = 78.71 marks

It is more conveninent to find the products of the scores and the frequencies by adding an extra column to the frequency table and the work as under:



Scores(x)        frequence(f)         (f.x)

    6 0                 1              60

    65                  2              130

    70                  6              420

    75                  8              600

    80                  7              560

    85                  5              425

    90                  3              270

    95                  2              190

   100                  1              100

   Sum                 35              2755

Mean=sum of the fx column/ sum of the f column  

             = 2755/35

             = 551 / 7

             = 78.71 marks

Find the value of x

The term x is an element of alphabets. Generally, in mathematics it is used as a variable. The variable is defined as the letter which represents a number (not only the letters, can a symbol also used as a variable). The number which is represented by the variable is called the co-efficient of that variable. In algebra, polynomials are formulated from one or more variables (such as x) and also by symbols.

Procedure of Solving for X:

State all the possibilities given.
Simplify the equation.
Isolate the variable to find its exact value
Simplify the equation to get the value of the variable.

Sample Problems:

Pro 1:  Solve the following equation and find the value of x.

                     2x + 5 = 3

Sol :    Given 2x + 5 = 3

   To simplify the equation subtract 5 on both sides

       2x + 5 - 5 = 3 – 5

        2x = -2

   Simplifying this, we get

          x = 1

Pro 2:  Solve the equation |2x + 5| + 3 = 5

Sol :   The equation to solve is given by.

            |2x + 5| + 3 = 5

   Subtract 3 to both sides of the equation and simplify.

               |2x + 5| = 2

   |2x + 5| is equal to 2 if 2x + 5 = 2.

   Simplify the above equation

            2x + 5 = 2.

   Isolate x and simplify

                    x = -3/2

Pro 3:   Given the system of equations
     x + y = 0
     2x + 3y = 2.

Solve and find the values of x and y.

Sol :   Multiply the first equation by 2.

         2x + 2y = 0

          2x + 3y = 2

   Solve the above equation by changing the sign of the terms in second equation.

              2x + 2y = 0

             -2x - 3y = -2

    which gives the solution

           -y = -2

    Therefore,    y = 2

   Substitute y value in first equation

        x + y = 0

        x + 2 = 0

    Hence,   x = -2

Pro 4:  Solve the equation 8x^4 + 3x^2 + 4x - 5x^-1 + 2x^-3 for x = 2.

Sol :    Evaluate the equation by substitute the value of x

            = 8x^4 + 3x^2 + 4x - 5x^-1 + 2x^-3

            = 8(2)4 + 3(2)2 + 4(2) - 5(2)-1 + 2(2)-3    

            = 8(16) + 12 + 8 - (5/2) + (1/4)

            = 128 +12 + 8 - 2.5 + 0.025

            = 145.525

Problem 4:

Pro 1:  Solve the quadratic equation x^2 + 5x + 6 = 0

Sol :   Given x^2 + 5x + 6 = 0

    For finding the roots of the equation, factorize it using trial and error method.

            x^2 + 5x + 6 = 0

             (x + 2) (x + 3) = 0

     Therefore the roots are

              x = -2 and x = -3