Thursday, February 28, 2013

Dividing Integers Solving Online

The natural number and negative numbers together with zero are called integers. The natural numbers are 1, 2, 3, 4….., the whole numbers are 0, 1, 2, 3… and the negative numbers are -1,-2,-3…. Therefore the set of integer can be -4,-3,-2,0,1,4,6……….. Here we have to learn about how to solve and divide the integers in online and its operations.

dividing integers solving online

Solving methods dividing integers online:


The operation of dividing integers in an online performed by four different ways, they are following,

Positive integer divided by positive integer = positive integer.
Negative integer divided by negative integer = positive integer.
Positive integer divided by negative integer = negative integer.
Negative integer divided by positive integer = negative integer.

Problems of Dividing Integers Solving Online :


Online example: 1

To solve the following integer 8/4

Solution:

Here the both numerator and denominator are positive.

So, = 8 / 4

=2

So the result is positive =+2

Example: 2

To solve following 66/9

Sol: Here also both numerator and denominator are positive so the answer will be positive

=divide both numerator and denominator by 3

=22/3

Answer is = +22/3

Example: 3

To the following, -12 divide by -6

Solution:

Given both integers are negative so the answer will be positive

Both numerator and denominator by -6

=2

Answer is positive =+2

Example: 4

To solve the following -76 divide by -4

Solution:

Both numerator and denominator are negative so the result is positive

Divide -4 by numerator and denominator

= 19

Answer is positive = +19

Example: 5

To solve the following integers

25 divide by -5

Solution:

Here the numerator is positive and denominator is negative so the answer will be negative

Divide both numerator and denominator by 5

=-5

The answer is negative = -5

Example: 6

Divide the following,-81 divide by 9

Solution:

Here the numerator is negative and denominator is positive so the answer will be negative

= divide both numerator and denominator by 9

= -9

The answer is negative = -9

Example: 7

-121 divide by 11

Solution:

Here numerator is negative and denominator is positive

= divide by 11

= -11

The answer is negative =-11

So in these solving and dividing integers online if any one that is numerator or denominator will be negative means the answer will be negative.

Similarly both are positive are negative the answer will be positive.

Wednesday, February 27, 2013

Geometric Distributions

Introduction:

Let we will discuss about the geometric distributions. The geometric distributions should be either of two discrete probability distributions in statistics and probability theory. The probability allocation of number X of Bernoulli trials desired to obtain one success, beard on the set { 1, 2, 3, ...}. The probability allocation of number Y = X − 1 of failures previous to the first success, maintained on the set { 0, 1, 2, 3, ... }  


More about geometric distributions:


  • The two distinct geometric distributions does not mystified with each other.
  • Most commonly, name shifted geometric allocation is accepted for the former one.
  • But, to keep away from ambiguity, it is measured wise to point out which is planned, by mentioning the range explicitly.
  • If the probability of success on every check should be p, then the probability that kth check is the first success is,
\Pr(X = k) = (1 - p)^{k-1}\,p\,

Where, k = 1, 2, 3, ....
  • Consistently, if the probability of success on each check is p, then the probability that there are k failures before the first success is
\Pr(Y=k) = (1 - p)^k\,p\,
Where, k = 0, 1, 2, 3, ....

  • In both case, the series of probabilities is a geometric series.

Example:

Assume a normal die is thrown frequently until the first time a "1" appears. The probability allocation of the number of times it is thrown is holded on the endless set { 1, 2, 3, ... }. They should have a geometric allocation with p = 1/6.

Moments and cumulants:
  • The predictable value of a geometrically allocated random variable X is 1/p , variance is (1 − p)/p2
\mathrm{E}(X) = \frac{1}{p},  \qquad\mathrm{var}(X) = \frac{1-p}{p^2}.

  • Likewise, the expected value of the geometrically dispersed random variable Y is (1 − p)/p, and its variance is (1 − p)/p2
\mathrm{E}(Y) = \frac{1-p}{p},  \qquad\mathrm{var}(Y) = \frac{1-p}{p^2}.
  • Let μ = (1 − p)/p be the expected value of Y and then the cumulants κn of the probability distributions of Y satisfy the recursion
\kappa_{n+1} = \mu(\mu+1) \frac{d\kappa_n}{d\mu}.

Outline of proof:

That the expected value is (1 − p)/p can be shown in the following way. Let Y be as above. Then

\begin{align} \mathrm{E}(Y) & {} =\sum_{k=0}^\infty (1-p)^k p\cdot k \\ & {} =p\sum_{k=0}^\infty(1-p)^k k \\ & {} = p\left[\frac{d}{dp}\left(-\sum_{k=0}^\infty (1-p)^k\right)\right](1-p) \\ & {} =-p(1-p)\frac{d}{dp}\frac{1}{p}=\frac{1-p}{p}. \end{align}

Tuesday, February 26, 2013

Percentage Change Calculator

A percentage change is a way to express a change in a variable. It represents the relative change between the old value and the new one.

The formula used to calculate the percentage change is

Percentage change = `((V2 - V1) / (V1)) * 100` .

V1- represents the old value

V2 - the new one.                                                              Source: - Wikipedia



some percentages as fractions:

a.       `5%=5/100=1/20` .

b.      `10%=10/100=1/10` .

c.      `25%=25/100=1/4` .

d.     `75%=75/100=3/4` .

e.      `125%=125/100=5/4` .

f.        `175%=175/100=7/4` .

g.       `(3 1/8)%=25/800=1/32` .

h.    ` (6 1/4)%=25/400=1/16` .

i.        `(8 1/3)%=25/300=1/12` .

j.        `(16 2/3)%=50/300=1/6` .

k.       `(66 2/3)%=200/300=2/3` .

l.        `(87 1/2)%=175/200=7/8` .

Calculation of Percentage:


Calculation of percentage:

The percent symbol can be treated as being equivalent to the pure number constant `1/100=0.01,`  while performing calculations with percentage.

If a number is first changed by`P% ` and then changed by `Q%` , then the net change in the number `=[P+Q+((PQ)/100)]` . Remember that any decreasing value in the formula should be taken as ‘negative’ and increasing value should be taken as ‘positive’.

Similarly, if A’s salary is `P%`  less than B’s salary, then the percentage by which B’s salary is more than A’s salary is`(100P)/(100-P)` .

If expenditure also, then percentage change in expenditure or revenue`=[P+Q+((PQ)/100)]` . Where ‘P’ is the percentage change in price and ‘Q’ is the percentage change in consumption.


Percentage change calculator - Example problems:


Percentage change calculator - Problem 1:-

Ram scored 86 runs in the cricket match on  Monday.  On Friday he scored 95 runs.  Calculate the Percentage of change?

Solution:-

Given

V2 = new value = 95 runs.

V1 = old value = 86 runs.

Percentage of change =  ?.

The formula used to calculate the percentage change is

Percentage change = `((V2 - V1) / (V1)) * 100` .

By plugging in the given values in to the formula we get

Percentage change  =` (95 - 86) / (86) * 100` .

The difference between  95 and 86 is  9.

By plugging in it to the formula we get the answer as

=`9 / 86` * 100.

The fraction 9/ 86 gives us 0.1046.

=0.1046 * 100

=10.46

The percentage of change is 10.46.

Percentage change calculator - Problem 2:-

Mary bought 40 Compact disks last month.  He bought only 30 this year.  Calculate is the percent of change.

Solution:-

Given :-

Here,

V2 = new value =  35 Compact disks

V1 = old value = 40 Compact disks

Percentage of change =?

The formula used to calculate the percentage change is

Percentage change = `((V2 - V1) / (V1)) * 100` .

V2 = new value.

V1 = old value

By plugging in the given values in to the formula we get

Percentage change =  `(35 -40) / (40)` * 100.

The difference between 35 and 40 is 5

By plugging in the given values in to the formula we get,
33333
= `5 / 40` * 100

The fraction 8/ 40 gives us  1/5.

`= 1/ 5 ` * 100.

= 20%

The percentage of change is 20 %


Percentage change calculator - Practice Problem:


Ex1 : John scored 16 runs in the cricket match on  Monday.  On Friday he scored 32 runs.  Calculate the Percentage of change?

Answer:-

The percentage of change is 50%

Ex2 : Calculate `40%`  of `625` .

Sol: `40% ` of a number `=2/5`  of the number `=2/5`  of `625=(2/5)(625)=250` .

Monday, February 25, 2013

Temperature Conversion

In this page we are going to discuss about temperature Conversion concept . Measurement is one of the important terms in day to day life. Temperature is usually measured in terms of Fahrenheit and Celsius. Temperature of is generally in terms of these two names.

Fahrenheit:

The degree Fahrenheit is usually represented as (F). Fahrenheit is names after the German scientist Gabriel Fahrenheit, who invented the Fahrenheit measurement. The zero degree in the Fahrenheit scale represents the lowest temperature recording.

Celsius:

The degree Celsius is usually represented as (C). Celsius is names after the Swedish astronomer Ander Celsius, who proposed the Celsius first. In Celsius temperature scale, water freezing point is given as 0 degrees and the boiling point of water is 100 degrees at standard atmospheric pressure.

Formula for Fahrenheit and Celsius Conversion


Fahrenheit to Celsius Conversion Formula:

The formula for converting Fahrenheit to Celsius conversion is given as,

Tc = (5/9)*(Tf-32)

Where,

Tc = temperature in degrees Celsius,

Tf = temperature in degrees



Celsius to Fahrenheit Conversion Formula:

The formula for converting Celsius to Fahrenheit conversion is given as,

Tf = (9/5)*Tc+32

Where,

Tc = temperature in degrees Celsius,

Tf = temperature in degrees Fahrenheit.


Examples on temperature Conversion

Below are the examples on fahrenheit to celsius conversion problems :

Example 1 : Convert 68 degree Fahrenheit to degree Celsius.

Solution:


The formula for converting Fahrenheit to Celsius conversion is,

Tc= (5/9)*(Tf-32)

Tc= 68

Tc= (5/9) * (68 – 32)

Tc= (5/9) * 36

Dividing 36 by 9, we get 4,

Tc= 5 * 4

Tc= 20 degree.

The answer is 20 degree Celsius

Example 2 :Convert 132 degree Fahrenheit to degree Celsius.

Solution:


The formula for converting Fahrenheit to Celsius conversion is,

Tc= (5/9)*(Tf-32)

Tf = 132

Tc= (5/9) * (132 – 32)

Tc= (5/9) * 100

Dividing 100 by 9, we get 11.11,

Tc= 5 * 11.11

Tc= 55.11 degree.

The answer is 55.11 degree Celsius.



Now see the Celsius to Fahrenheit conversion problems:

Example 1 :  Convert 50 degree Celsius to degree Fahrenheit.

Solution:


The formula for converting Celsius to Fahrenheit is given as,

Tf= (9/5)*Tc+32

Tf= (9/5) * 50 + 32

Tf= 9 * 10 + 32

Multiplying 9 and 10 we get 90,

Tf= 90 + 32

Tf= 122 degree Fahrenheit

The answer is 122 degree Fahrenheit.


Example 2 :  Convert 42 degree Celsius to degree Fahrenheit.

Solution:


The formula for converting Celsius to Fahrenheit is given as,

Tf= (9/5)*Tc+32

Tf= (9/5) * 42 + 32

Tf= 9 * 8.4 + 32

Multiplying 9 and 8.4 we get 75.6,

Tf= 75.6 + 32

Tf= 107.6 degree Fahrenheit.

The answer is 107.6 degree Fahrenheit.

Friday, February 22, 2013

Introduction to Probability and Statistics

Introduction for Probability:

Introduction for Probability is the possibility that rather will happen - how to be expected it is that some event will happen. Now and again you can measure a probability with a number: "10% chance of rain", or you know how to use words such as impossible, unlikely, and possible, even chance, likely and certain. when a coin is tossed there is a probability of getting head and tail possible outcomes for an experiment occur is sample space.

Example: "our team may won match today"

Here is a probability formula:

`P(A) = (The Number Of Ways Event A Can Occur)/(The Total Number of possibLe outcomes)`

Introduction to Statistics:

Statistics is the branch which is applied by mathematics, which deals with the scientific analysis of data. The word ‘Statistics’ is derived from Latin word ‘Status’ which means ‘political state’. In statistics introduction datas are in two types which are primary and secondary datas. Sometimes an investigator uses the primary data of another investigator collected for a different purpose. Such data are called statistics secondary data.

From the data we have learnt to calculate the measures of the central tendency like mean, median and mode. These central measures do not give us all the details about the distribution.Further descriptions of the data called measures of dispersion are necessary. According to A.L. Bowley, “Dispersion is the measure of the variation of the individual item”. That is the dispersion is used to indicate the extent to which the data is spread.

Statistics properties:

Statistics deals with three properties, which are  Mean, Median , Mode

Examples:
A cancer patient wants to identify the probability that he will survive for at least 5 years. By collecting data on survival rates of people in a similar situation, it is possible to obtain an empirical estimate of survival rates. We cannot know whether or not the patient will survive, or even know exactly what the probability of survival is. However, we can estimate the proportion of patients who survive from data.

Thursday, February 21, 2013

Common Pythagorean Triples


Introduction to Pythagoras online study:

The Pythagoras Theorem is a statement relating the lengths of the sides of any right triangle.

 The theorem states that:

For any right triangle, the square of the hypotenuse
is equal to the sum of the squares of the other two sides.

Mathematically, this is written:

c^2 = a^2 + b^2

We define the side of the triangle opposite from the right angle to be the hypotenuse, c. It is the longest side of the three sides of the right triangle. The other two sides are labelled as a and b.

pythagoras theorem



Pythagoras generalized the result to any right triangle. There are many different algebraic and geometric proofs of the theorem. Most of these begin with a construction of squares on a sketch of a basic right triangle. We show squares drawn on the three sides of the triangle. For a square with a side equal to a, the area is given by:

A = a * a = a2

So the Pythagorean theorem states the area c2 of the square drawn on the hypotenuse is equal to the area a2 of the square drawn on side a plus the area b2 of the square drawn on side b.

pythagoras online study-Pythagorean triplets


A knowledge of Pythagorean triplets will also help the student in working the problems at a faster pace.

 The study of these Pythagorean triples began long before the time of Pythagoras.

There are Babylonian tablets that contain lists of such triples, including quite large ones.

There are many Pythagorean triangles all of whose sides are natural numbers. The most famous has sides 3, 4,

and 5. Here are the first few examples:

32 + 42 = 52;

52 + 122 = 132;

82 + 152 = 172;

282 + 452 = 532

There are infinitely many Pythagorean triples,that is triples of natural numbers (a; b; c) satisfying the equation a2 + b2 = c2.

If we take a Pythagorean triple (a; b; c),and multiply it by some other number d, then we obtain a new Pythagorean triple

(da; db; dc). This is true because,

(da)2 + (db)2 = d2(a2 + b2) = d2c2 = (dc)2 :

Clearly these new Pythagorean triples are not very interesting. So we will concentrate our attention on triples with no common factors.They are primitive Pythagorean triples

A primitive Pythagorean triple (or PPT for short) is a triple of numbers

(a; b; c) so that a, b, and c have no common factors1 and satisfy

a2 + b2 = c2:

There are 16 primitive Pythagorean triples with c ≤ 100:

( 3 , 4 , 5 )

( 5, 12, 13)

( 7, 24, 25)

( 8, 15, 17)

( 9, 40, 41

(11, 60, 61)

(12, 35, 37)

(13, 84, 85)

(16, 63, 65)

(20, 21, 29)

(28, 45, 53)

(33, 56, 65)

(36, 77, 85)

(39, 80, 89)

(48, 55, 73)

(65, 72, 97)

 One interesting observation in a primitive  Pythagoras triple is  either a or b must be a multiple of 3.


pythagoras online study-Solved examples


The Pythagorean Theorem must work in any 90 degree triangle. This means that if you know two of the sides, you can always find the third one.

 pythagoras solution1



In the right triangle, we know that:

c^2 = 6^2 + 8^2

Simplifying the squares gives:

                                                   c2= 36 + 64

                                                  c2 = 100    

                                                   c = 10       

                                      (taking the square root of 100)



In this example, the missing side is not the long one. But the theorem still works, as long as you start with the hypotenuse:

pythagoras solution2

                                                15^2 = a^2 + 9^2

Simplifying the squares gives:

                                                225 = a2 + 81

                                         225 - 81 = a2                 

                                                144 = a2        

                                                  12 = a  

                                                  a   = 12

                              (Notice that we had to rearrange the equation)

Wednesday, February 20, 2013

Learn Intercept Formula

Learn Intercept formula is nothing but slope intercept formula. Before going to learn intercept formula we need to know why it is called so.

It is called slope intercept form because the equation includes slope and the y-intercept. So now we know the reason why it is called slope intercept form. Now coming to actual concept.....

The general form of slope intercept form is:

y= mx^+b

Where m--> slope of the line.

b --> y-intercept.

y --> y-coordinate.

x -->x-coordinate.

Formula - learn intercept formula


Slope intercept form is the simplest of all forms as we just need to plug in the values of slope(m) and y-intercept(b).

Now how are we going to get the slope??. In problems the slope might be mentioned directly or two points through which the line passes might be given. When the second case occurs slope can be find out using formula
slope(m)= (change in x)/(change in y)

Now let us consider that two points are (x1,y1) and (x2,y2)  then slope is given by

m=(y2-y1)/(x2-x1)

We have learnt how to find slope in the slope intercept form. now the next one to be calculated is y-intercept(b).
For this we need to plug the point through which the line passes.

This point will be mentioned in the question. If slope is m and the point is (p,q) then

plugging these values in slope intercept form we get

q= m*p+b

==> b= q-mp.

Now we have slope and y-intercept substituting these values we get slope intercept form.

I think to learn intercept form is very easy  and... cool

Examples on learn intercept formula


Ex1:  What is slope intercept form of line having slope 2 and y-intercept 3?
Sol1:

Given slope (m)=2, and y-intercept (b)=3

Plug these values in slope intercept form y=mx+b

Then        y= 2*x+3

y=2x+3.

So slope intercept form of a line having slope 2 and y-intercept 3 is

y=2x+3

Ex2:  What is slope intercept form of line passing through the point (2, 3) and having a slope of 4.
Sol 2:

Here we have slope = 4 and y-intercept is not mentioned but the point through which line passes is given.

Plug m=4 and (x,y)= (2,3) in the slope intercept form y=mx+b.

Then we get

3=4*2+b.

==>b+8 =3

==>  b=3-8=-5

So y-intercept is b=-5 and we have slope as m=4.

Plugging these in slope intercept form we get

y= 4x -5.

Ex3:  Express the equation 3x+4y+5=0 in slope intercept form.
Sol 3:

The given equation is 3x+4y+5=0.

In order to convert it in to slope intercept form bring y terms on one side and the remaining terms to other side.

For this subtract with 3x and 5 on both sides

3x+4y+5-3x-5=-3x -5

By doing this 3x and 5 get cancelled and the equation becomes

4y= -3x-5

Divide both sides by 4

4y /4 = (-3x-5)/4

y=-(3/4)x  - 5/4

Thus the slope intercept form of 3x+4y+5=0 is y=-(3/4)x  - 5/4

Monday, February 18, 2013

Mean Value Theorem Derivatives

In calculus, the mean value theorem states, roughly, that given an arc of a smooth continuous (derivatives) curve, there is at least one point on that arc at which the derivative (slope) of the curve is equal (parallel) to the "average" derivative of the arc. It is used to prove theorems that make global conclusions about a function on an interval starting from local hypotheses about derivatives at points of the interval. I like to share this Anti derivative with you all through my article.

Formal Statement of Mean Value Theorem Derivatives:

Let f: [x, y] → R be a continuous function on the closed interval [x, y] , and differentiable on the open interval (x, y), where x < y. Then there exists some z in (x, y) such that,

f'(z) = `(f(y) - f(x))/(y - x)`

The mean value theorem is a generalization of Rolle's Theorem, which assumes f(x) = f (y), so that the right-hand side above is zero.

Only one needs to assume that f: [x, y] → R is continuous on [x, y] , and that for every m in (x, y) the limit,

`lim_(h ->0) (f(m + h) - f(m))/(h)`

exists as a finite number or equals + ∞ or − ∞. If finite, that limit equals f′(m). An example where this version of the theorem applies is given by the real-valued cube root function mapping m to m1/3, whose derivative tends to infinity at the origin.

Note that the theorem is false if a differentiable function is complex-valued instead of real-valued.

For example, define f(m) = eim for all real m. Then

f (2π) − f(0) = 0 = 0(2π − 0)

while, |f′(m)| = 1.

- Source Wikipedia


Proof of Mean Value Theorem Derivatives:


Let g(m) = f(m) − rm, where r is a constant, because f is continuous on the closed interval [x, y] and differentiable on the open interval (x, y), now we want to select r, so as g satisfies the conditions of the Theorem,

g (x) = g (y) `hArr` f (x) − rx =  f (y) − ry

`hArr` ry − rx = f (y) − f(x)

`hArr` r(y − x) = f (y) − f(x)

`hArr` r = `(f(y) - f(x))/(y - x)`

By Rolle's theorem, g is continuous on the closed interval [a, b] and g(a) = g(b), there is some c in (a, b) for which g′(c) = 0, and it follows from the equality g(x) = f(x) − rx that,

f' (c) = g' (c) + r = 0 + r =  `(f(b) - f(a))/(b - a)`

Thursday, February 14, 2013

Algebra 2 Help and Answers

Algebra 2 help us find the unknown quantities with the help of known quantities. In algebra, we frequently use letters to represent numbers. Algebra 2 help includes real numbers, complex numbers, vectors, matrices etc. Algebra 2 is the study of the rules of relations and operations, and the constructions arising from them. An algebraic expression represents a scale that gets added or subtracted or multiplied or divided on both sides. In algebra 2 help and answers numbers are considered as constants.


Algebra 2 help and answers example Questions:


The following problems gives different answers to algebra 2 problems.

Ex 1:  Determine all real solutions to the equation

Sqrt (2 x + 13) = x - 5

Sol :   Given

sqrt (2 x + 13) = x - 5

We raise both sides to power 2.
[Sqrt (2 x + 13)] 2 = (x - 5) 2

And simplify.
2x +13 = x 2 - 10 x +25

Write the equation with right side equation to 0.
X 2 - 8 x + 12 = 0

It is a quadratic equation with 2 solutions
x = 6 and x = 2

Ex 2 :  Determine all real solutions to the equation

Sqrt (x + 8) = 12

Sol :  Given

sqrt (x + 8) = 12

We raise both sides to power two(2) in order to clear the square root.
[Sqrt (x + 8)] 2 = 12 2

And simplify
x + 8 = 144

Solve for x.
x = 136

Ex 3:   Determine all real solutions to the equation

Sqrt (x 2 – 13x+72) = 6

Sol :   Given

Sqrt (x 2 – 13x+72) = 6

We raise both sides to power 2.
[Sqrt (x 2 – 13x+72)] 2 = (6) 2

And simplify.
X 2 – 13x+72= 36

Write the equation with right side equation to 0.
X 2 - 13 x + 36 = 0

It is a quadratic equation with 2 solutions
x = 9 and  x = 4


Algebra 2 help and answers practice problems:


1) Determine all real solutions to the equation

Sqrt (x + 25) = 13

Ans : x=12

2) Determine all real solutions to the equation

Sqrt (x 2 – 11x+55) = 5

Ans : x=5 and x=6

3) Determine all real solutions to the equation

Sqrt (2 x + 24) = x

Ans : x=6 and x= - 4

Wednesday, February 13, 2013

Complex Polygon

Complex polygon is a polygon whose sides cross over eachother one or more times. If the number of cross over increases, the complexity of the polygon also increases. Below are some of the complex polygons

Area of a Complex Polygon


There are three steps to find the area of the complex polygon. They are as follows:

Step 1: Break the polygon into simple rectangle, square or triangle.

Step 2: Find the area of all rectangle, square or triangle.

Step 3: Add all the area and get the area of complex polygon.

With the help of the above steps, we can easily find the area of any complex polygons.


Examples

Given below are some of the examples to find the area of a complex polygon.

Example 1:

Find the area of following polygon


DE= 5cm, EF =8cm, FD = 5 cm, AB = 14cm, CD = 10cm and the distance between these two parallel side AB and CD is 5cm

Solution:

Step 1: Break the polygon into triangle and trapezoid.


Here, the given complex polygon is braked into two simple polygon. It is separated at the point D

Step 2: Find the area of triangle (D1EF) and trapezoid (ABCD2).

Area of triangle (D1EF)

Value of s = (D1E + EF + FD1) / 2

Value of s = (5+8+5) / 2

Value of s = 9 cm

Area of triangle (D1EF) = SquareRoot (s(s - D1E)(s - EF)(s - FD1))

= SquareRoot (9(9 - 5)(9 - 8)(9 - 5))

= SquareRoot (9 x 4 x1 x 4)

=  SquareRoot (144)

= 12 square cm

Area of Trapezoid (ABCD2) =  ½ ( AB +CD2 ) x ( Perpendicular distance between AB  and  CD2 )

= ½ (10 +14) x 5

= ½ (24) x 5

= 12 x 5

= 60 square cm

Step 3: Add all the area of triangle and trapezoid to get area of complex polygon.

Area of Complex polygon = Area of triangle (D1EF) + Area of Trapezoid ABCD2)

= 12 square cm  +  60 square cm

= 72 square cm

Monday, February 11, 2013

Binomial probability formula

Binomial is deals with the polynomial that has two terms is known as a binomial e.g.5x + 3y, 2x^2 – 5xy.Probability is the arithmetical quantify of the possibility of an event to occur. If in an examine there are n feasible ways completely and mutually exclusive and out of them in m ways in the event A occur, it is given by (m / n) if in a random sequence of n trails of an events, M are favor to the event, then ratio is (M / n).by studying in both combined as binomial probability as P(A) = m / n.

The possibility of an event can be conveyed as a binomial probability if its conclusions can be wrecked down into two probability of p and q, where p and q are balancing (i.e. p + q = 1)


Binomial Probability Formula - Explained


The probability of getting exact value of k in n trials is given by using the formula,

`P(X = x) = ((n),(k)) p^k q^(n-k) `
where
` q= (1-p)`
or
`P(X = r) =nCr p^r(1-p)^(n-r)`

where
n = Number of events.
r = Number of successful events in the trials.
p = Probability of success in the single trial of the events.
nCr = `(( (n!) / (n-r)! ) / (r!))`
1-p = Probability of failure in the trial.


Examples on Binomial Probability Distribution

Example 1:

Assume that there are taking about 10 question multiple choice test. If each question has four choices and you have to guess on each question, what is the probability of getting exactly 7 questions correct using binomial Probability formula?


Solution:
General Formula for finding solution is,

`P(X = x) = ((n),(k)) p^k q^(n-k) `

n = 10
k = 7
n – k = 3
p = 0.25 = probability of guess the correct answer on a question
q = 0.75 = probability of guess the wrong answer on a question

P(7 Correct guesses out of 10 questions) = `((10),(7))` (0.25)7(0.75)3
? 0.0031 approximate value

Therefore if someone guesses 10 answers on a multiple choice test with 4 options, they have about a 5.8% chance of getting 5 and only 5 correct answers. If 5 or more correct answers are needed to pass then probability of passing can be calculated by adding the probability of getting 5 (and only 5) answers correct, 6 (and only 6) answers correct, and so on up to 10 answers correct. Total probability of  5 or more correct answer is approximate percentage is  7.8


Example 2:
Suppose a die is tossed 7 times. What is the probability of getting exactly 3 fours?

Solution:


General Formula for finding solution

`P(X = r) =nCr p^r(1-p)^(n-r)`

Step 1:
Number of trials n = 7
Number of success r =3

Probability of success in any single trial p is given as 1/6 or 0.167

Step 2:

To calculate nCr formula is used.

nCr = `(n!)/((n-r)!(r!))`

= `(7!)/((7-3)!(3!))`

= `(7!)/((4)!(3!))`

= `(5040)/((24)(6))`

= `(5040)/(144)`

= 35

Step 3:

Find pr.
pr =  0.1673
= 0.004657463

step 4:

To Find (1-p)n-r Calculate 1-p and n-r.
1-p = 1-0.167 = 0.833
n-r = 7-3 = 4

Step 5:

Find (1-p)n-r.
= 0.8334 = 0.481481944

Step 6:

Solve P(X = r) = nCr p r (1-p)n-r
= 10 × 0.004657463 × 0.481481944
0.0224248434

The probability of getting exactly 3 fours is 0.0224248434

Friday, February 8, 2013

Solving Online Venn Diagram of Quadrilaterals

We know that on combining three non-collinear points in pairs, the figure obtained as a triangle. Now, let us mark four points and see what we obtain on combining them in pairs in some order. Such a figure created by combining four points in an order is called quadrilateral. In a quadrilateral, a pair of opposite sides is parallel.

In online, students can learn about quadrilaterals. Through online, students can have interactive sessions with tutors. Online is one of the efficient tools for one to one learning. In this article, we are going to discuss about solving online venn diagram of quadrilaterals.

Solving Online Venn Diagram of Quadrilaterals - Definition and its Types:

Definition of Quadrilateral:


Quadrilateral is a geometrical figure that consists of four ends called verticals joined to each other by straight-line segments or sides called edges. The quadrilateral are classified depends on the length of side, angle and diagonals of object.

Solving online venn diagram of quadrilaterals - Types:

There are two types of Quadrilaterals in geometry. The following are the two types of Quadrilaterals.

Convex quadrilaterals:

Parallelograms
Square
Rectangle
Rhombus

Concave quadrilaterals:

Trapezium
Kite

Solving online venn diagram of quadrilaterals:
Venn diagram - Quadrilaterals


Solving Online Activities with Quadrilaterals - Descriptions and Example Problems:

Parallelogram:


Parallelogram is one of the classifications of quadrilateral
Parallelogram consists of parallel opposite sides
Parallelogram consists of equal length
Parallelogram consists of equal opposite angles
Parallelogram

Formula:

Area = b × h sq. units (b – Base, h - Height)

Example:

The base of a parallelogram is 19 and the height is 12. What is the area of this parallelogram?

Solution:

Area of parallelogram = b * h

= 19 * 12

= 228sq.units

Square:

Parallel lines are lines are exactly the same length and height that has never cross.

Square is a one of the classifications of quadrilateral
A square shape is same to both rhombus and rectangle
Square consists of equal parallel sides and also all angles are right angle (90°) only
The Square is a parallelogram having an angle, equal to right angle and adjacent sides equal.
Square

Formula:

Area = a2    (a - area)


Example:

Find the area of square, when side length is 22cm

Solution:


Area of square = a2

= 22 * 22

= 484 cm2

Rhombus properties:

Rhombus is a parallelogram having its adjacent sides equal but more of whose angles is a right angles.

Rhombus consists of four-sided geometrical shape and also all sides have same length
Rhombus consists of parallel opposite sides
It consists of equal opposite angles
It is also one of the parallelograms
Rhombus

Formula:

Area = `1 / 2` (d1 × d2) sq. units

Where, d1, d2 are diagonals.

Example:

Find the area of rhombus d1 = 20 cm, d2 = 29 cm

Solution:

Area of rhombus = `1 / 2` (d1 * d2)

= `1 / 2` (20 * 29)

= (10 * 29)

= 290 cm2

Trapezium:

A quadrilateral should have at least one pair of parallel sides. The quadrilateral one side of opposite sides are parallel if non parallel of opposite sides of a trapezoid are congruent it is called as isosceles triangle.

Trapezium is also otherwise known as trapezoid
Trapezium is one of the main types of quadrilateral
Trapezium has one pair of parallel that present in opposite sides
Trapezoid

Formula:

Area = `1 / 2` (a + b) × h sq. units.


Example:

Find the area of trapezoid height = 11 cm, a = 20 cm, b = 29 cm

Solution:

Area of trapezoid = `1 / 2 ` (a + b)* h

=` 1 / 2` (20 + 29) * 11

= `1 / 2 ` (49) * 11

= 24.5 * 11

= 269.5 square cm



Thursday, February 7, 2013

Right Triangle Similarity Theorem

The right triangle similarity theorem is otherwise called as Pythagoras theorem. This right angle theorem was introduced by Philosopher and Greek Mathematician, Pythagoras. The right triangle is a part of geometrical figures. This theorem is used for finding the length of any sides of a right triangle. The right triangles are special triangles that contain only one right angle. Here the right triangle is a triangle that measuring an angle `90^o`

Statement for Right Triangle Theorem:

In terms of area, we can define this theorem. In any right triangle, the area of the square whose side is hypotenuse is equal to the sum of the areas of the other two sides. Here the hypotenuse refers to the side that opposite to the right triangle. The other two sides of a triangle meet at right angle.

tri

Here Hypotenuse is the side opposite to the right angle. an adjacent side is the side adjacent to the given angle and the opposite side is the side that opposite to given  .

According to this theorem, the equation can be given as

(Opposite)2+ (Adjacent)2 = (Hypotenuse)2

Here, an angle  value lies between 0 and 90 degree. Here, 90 degree is one of the interior angle and the other two interior angles are complementary. Complementary means the angle value less than 90 degree. The other two interior angle sum should be equal to 90 degree.  This right triangle similarity theorem can also be used in trigonometric functions. This right triangle forms six possible ratios in trigonometry.



Other Forms of the Equation:
tri1


Here consider a is opposite side, b is an adjacent side and c is hypotenuse. Therefore, an equation can be written as a2 + b2 = c2. To find c, an equation can be written as c= `sqrt(a^(2)+b^(2))` . If c is known value, the length of one side is given; the following equations can be used;

a =`sqrt(c^(2)+b^(2))`   (or)       b = `sqrt(c^(2)-c^(2))`

Proof:

This theorem proof is based on the proportionality of two similar triangles. It depends on the ratio of any two corresponding sides of similar triangles.

trian

Let ABC represent a right triangle, with the right angle located at B. Here, H is the altitude drawn from B and it also intersects AC. The point H divides the hypotenuse c into two parts d and e. The new triangle ABH is similar to the triangle ABC because both the triangles have a right angle. Similarly, the triangle BCH is also similar to the triangle ABC. Thus the proof of similarity of the right triangles requires the Triangle postulate i.e., the sum of the angles in a triangle is 2 right angles, and is equivalent to the parallel postulate. Similarity of the right angle triangles leads to the equality of ratios of their corresponding sides:

a/c =e/a and b/c= d/b

In this the first result is equal to cosine angle and the second result is equal to sine angle.

These ratios can be written as:

a2= c*e and b2=c*d

Summing these two equations we get:

a2 + b2 = (c*e) + (c*d) = c* (e + d) = c*c = c2

Therefore a2+ b2 = c2

Hence, right triangle similarity theorem is proved.

Wednesday, February 6, 2013

Graphing Calculator Polar Coordinates

In this article, we are going to study about graphing calculator polar coordinates. It is very simple to work on the polar coordinates graphing calculator. For example, if we want to graphing radians r = 3 and `theta=45^o` on the polar graph paper, we have to just enter `3` in the radians text box and `45^o` in angles in degrees text box and then click the enter button, Now the calculator automatically plot the polar coordinates on the display of the polar graph paper. These process are shown in the below figure.

Graphing calculator polar coordinates


Graphing Calculator Polar Coordinates - Example 1:

Study and graphing of the given polar coordinates.

(2, 50o), (3, 210o), and (-2, 135o).

Solution:

Graphing the given polar coordinates are on the below polar graph paper:

Graphing calculator polar coordinates

Steps for plotting polar coordinates (2, 50o):

1:   The radian 2 is marked from the origin (pole) on the positive side of the x axis.

2:   The given angle 50o is rotated in anticlockwise directions and then plot the required polar coordinates (2, 50o) as shown in figure.

Steps for plotting polar coordinates (3, 210o):

1:   The radian 3 is marked from the origin (pole) on the positive side of the x axis.

2:   The given angle 210o is rotated in counterclockwise directions and then plot the polar coordinate (3, 210o) as shown in figure.

Steps for plotting negative polar coordinates (-2, 135o):

1:   The radian 2 is marked from the origin on the positive side of the x axis.

2:   The given angle 135o , is rotated in anticlockwise directions and it is determined.

3:   Now we can extend the 135o degree line in opposite direction (315o) where we get  the radian 2 in negative sign and then plot the polar coordinates (-2, 135o).

Graphing Calculator Polar Coordinates - Example 2:

Study and graphing of the given polar coordinates by using the above calculator.

(2.5, 60o), (-1, 75o), and (3, -30o).

Solution:
Polar coordinates are on the below polar graph paper:

Graphing calculator polar coordinates

Steps for plotting polar coordinates (2.5, 60o):

Step 1:   The radian 2.5 is marked from the origin (pole) on the positive side of the x axis.

Step 2:   The given angle 60o is rotated in anticlockwise directions and then plot the required polar coordinates (2,.5 60o) as shown in figure.

Steps for plotting polar coordinates (3, -30o):

Step 1:   The radian 3 is marked from the origin (pole) on the positive side of the x axis.

Step 2:   The given angle 30o is rotated in clockwise directions and then plot the polar coordinate (3, -30o) as shown in figure.

Steps for plotting negative polar coordinates (-1, 75o):

Step 1:   The radian 1 is marked from the origin on the positive side of the x axis.

Step 2:   The given angle 75o , is rotated in anticlockwise directions and it is determined.

Step 3:   Now we can extend the 75o degree line in opposite direction (255o) where we get  the radian 1 in negative sign and then plot the polar coordinates (-1, 75o).

Monday, February 4, 2013

Definition Non Linear

In math, a nonlinear system is one of the system which is not linear, that is, a system which does not assure the superposition principle, or whose result is not directly proportional to its input. The equation used to solve the non linear problem is f(x)=C. This function is called as linear if f(x) is a non linear. This article gives some of the examples about the non linear by using this definition.


Examples of Definition Non Linear:

By using the definition of non linear solve the following examples.

Example problem1:

Use the following function rule to find f(7).

`f(x)=2x^2`

Solution:

Given `f(x)=2x^2` ; Find f(7).

Plug x = 7 into the function and simplify.

`f(x) = 2x^2`

`f(7) = 2(7)^2`            Plug in x = 7

f(7) = 2(49)                  Square

f(7) = 98                       Multiply

Which is the required solution.

Example problem2:

Use the following function rule to find f(8).

`f(x) = (x - 4)^2`

Solution:


Given:

`f(x) = (x - 4)^2`

Find: f(8)

Plug x = 8 into the function and simplify.

`f(x) =(x - 4)^2`

`f(8) = ( 8 - 4)^2`            Plug in x = 8

`f(8) = (4)^2`                  Subtract

f(8) = 16                          Square

Which is the required solution.

Example problem3:

Use the following function rule to find f(1).

f(x) = 5 – 4|x|

Solution:

Given: f(x) = 5 – 4|x|; Find f(1)

Plug x = 1 into the function and simplify.

f(x) = 5 –4|x|

f(1) =5 – 4|1|                  Plug in x = 1

f(1) =5 – 4(1)                 Take the absolute value

f(1) =5 – 4                       Multiply

f(1) = 1                            Subtract

Which is the required solution.

Practice Problems of Definition Non Linear:

Problem 1:

Use the following function rule to find f(7).

f(x) = -9|x| + 4

Solution:

f(7) = -59

Problem 2:

Use the following function rule to find f(-8).

`f(x) = 3x^2`

Solution:

f(-8) = 192

Friday, February 1, 2013

Relevance and Application of Exponential Functions in Real Life Situations

In this article relevance and application of exponential functions in real life situations , we will see how exponential function used in real life. Exponential function is used in more real life application like compound interest, problem based on population, problem based on radioactive decay, mortgage problems. Let us work out some problem to make understand that application of an exponential function.

Exponential Function Growth:

Exponential function growth`g=c(p)^t where,`

c-Number at initial

p-growth factor (p =1+r,here r is given growth percentage)

t-time

Example 1 -  Relevance and application of exponential functions in real life situations

A group of 1000 people increase by 5% in an hour near to accident place. How many people will be in the crowd after 3 hour?

Given:

c=1000

p =1+r =1+0.05=1.05

t =3 hour

Solution:

Exponential function growth g=c(p)^t

Substitute the given data in the formulae

g=1000(1.05)^3

g=1000(1.1576)

g=1157.6

g=1158 people

Exponential Function Decay:

Exponential function decay `d=c(p)^t`

where,

c-Number at initial

p-growth factor

(p=1-r,here r is given decay percentage)

t-time

Example 2: Relevance and application of exponential functions in real life situations

The price of a violin is  $1,000 which decreases at a rate of interest of 3%.What is the price of a violin after 2 year?

Given:

c=1,000

p=1-r=1-0.03=0.97

t=2 year

Solution:

Exponential function decay `d=c(p)^t`

`d=1,000(0.97)^2`

d=1,000(0.9409)

d=940.9

d=941

The price of a violin after 2 year=$941

Example 3: Relevance and application of exponential functions in real life situations

John invests $50,000 for 2 year with the interest of 4% compounded half yearly .Find out the compound interest for his investment at a given rate of interest?

Given:

P=$50,000

R=4%

N=2 year

Solution:

Compound interest =` P[1+(R/2)/100]^(2n)`

`=50,000[1+(4/2)/(100)]4`

`=50,000[1+(2/100)]4`

`=50,000xx(102/100)xx(102/100) xx(102/100) xx(102/100)`

=$50,000(1.0824)

=$54121.6

Compound interest=Total amount-principle

=54,121.6-50,000

=$4,121.6

Example 4: Relevance and application of exponential functions in real life situations

Calculate how long blood clotting cell will take to produce 20,400

Solution:

We need to consider  assume function to calculate population based on problems

`f(t)=2^t`

`20,400=2^t`

Find the natural logarithm on both side

`ln(20,400)=ln(2^t)`

ln(20,400)= t ln(2)

`t= ln(20,400)/ ln(2)`

`t=(9.923)/0.693=14.29`

After 14.32 min the blood clotting cell can produce 20,400  cell.