Right Circular Cylinders

Let us study about Right Circular Cylinders,
A prism shaped solid whose bases are circles is a cylinder. If the segment joining the centers of the circles of a cylinder is perpendicular to the planes of the bases, the cylinder is a right circular cylinder. In Figure 1, cylinder (a) is a right circular cylinder and cylinder (b) is an oblique circular cylinder.

Figure 1 Different types of circular cylinders.

Lateral area, total area, and volume for right circular cylinders are found in the same way as they are for right prisms.

If a cylinder is pictured as a soup can, its lateral area is the area of the label. If the label is carefully peeled off, the label becomes a rectangle, as shown in Figure 2

Figure 2 The lateral area of a cylinder.

The area of the label is the area of a rectangle with a height the same as the altitude of the can and a base the same as the circumference of the lid of the can.

Theorem: The lateral area, LA, of a right circular cylinder with a base circumference C and an altitude h is given by the following equation.




Theorem: The total area, TA, of a right circular cylinder with lateral area LA and a base area B is given by the following equation.

Theorem: The volume of a right circular cylinder, V, with a base area B and altitude h is given by the following equation.

Example 1: Figure 3 is a right circular cylinder; find (a) LA (b) TA and (c) V.

Figure 3

Finding the lateral area, total area, and volume of a right circular cylinder.

Hope the above explanation helped you.

Rectangle

Introduction:
It is very easy to identify a Rectangle,it is a square figure we can find many objects around us of this shape.
A rectangle is a four-sided polygon and a flat shape with straight sides. Here every angle is a right angle (90 degree). And also opposite sides of rectangles are equal length and parallel.

A crossed rectangle is a complex rectangles, also called a butterfly rectangle or bow-tie rectangle .The rectangles are used to many episodic tessellation patterns, in stonework.



Properties and Formula of rectangles:

A simple rectangle has the following Properties:

* Rectangle is an isogonal..
* In rectangles, the two diagonals are equal in length.
* And also opposite sides are equal in length.
* Rectangle has two lines of rotating symmetry and reflection symmetry of order two.
* The double polygon of a rectangle is a rhombus. It is cyclic and convex.
* All angles are 90 degrees.
* The two diagonals bisect each other.
* Opposite sides are parallel.

Formula:

If a simple rectangle has length l and width w,

* Area A = lw,
* Perimeter P = 2l + 2w = 2(l + w),
* Each diagonal has length √l2+w2
* And when l = w, the rectangle is a square.

Examples of rectangles:

Example 1:

Find the area of rectangles with the given length 5 cm and width 6cm.

Solution:

Area = l * w

= 5 * 6

= 30 cm2

Answer: 30 cm2

Example 2:

Find the Perimeter and Area of rectangles with the given length 8 cm and width 10 cm.

Solution:

Perimeter P = 2(l + w)

Here we can add the values of length and width and multiplied by 2.

=2(8+10)

=2(18)

=36 cm

Area A = l * w

= 8 * 10

= 80 cm2

Answer: 80 cm2

Hope you like the above example of Rectangle.Please leave your comments, if you have any doubts.

Grouped Frequency

Grouped Frequency:

Before we get into the details of what is grouped frequency,let us first learn about the meaning of frequency.

What is the meaning of Frequency??

A frequency distribution is a tabular collection of data showing the frequency of each observation.

Frequency distributions are two types:

* Discrete frequency distribution
* Grouped frequency distribution

Grouped Frequency:


The grouped frequency table is the frequency data value that occur the number of times in the frequency table. It is the particular value that occur number of times in the grouped frequency table.

For example if five students have scored 90 marks in mathematics in a test

Then the mark scored 90 is the frequency of 5 members. Here f can be represented as frequency of grouped frequency table.

The frequency table can be constructed by arranging collected data value. The frequency tables are arranged in ascending order by magnitude with their corresponding frequencies.Now let us look at few examples of Grouped Frequency.

Example of Grouped Frequency Table:

Grouped frequency distribution

If the number of observation is large and the difference between the greatest and the smallest observations is large, then condense the data into classes or groups

There are two methods classifying the data according to the class interval:

* Exclusive method
* Inclusive method

Methods of Classifying Data:

Exclusive method:

when the class intervals are so fixed that the upper limit of one class is the lower limit of the next class, it is called Exclusive Method of classification. In this method upper limit of the class is not included in the class.

Inclusive Method:

In this method the classes are so formed that the upper limit of a class is included in that class.
Hope you like the above example of Grouped Frequency.Please leave your comments, if you have any doubts.

Pie Chart:

Pie Chart:

A circle is divided by several radii into sectors whose relative areas represent the relative magnitudes of quantities or the relative frequencies of items in a frequency distribution.The basic understanding of a Pie Chart is A pie chart (or a circle graph) is a circular chart divided into sectors, illustrating proportion. In a pie chart, the arc length of each sector (and consequently its central angle and area), is proportional to the quantity it represents. When angles are measured with 1 turn as unit then a number of percent is identified with the same number of centiturns. Together, the sectors create a full disk. It is named for its resemblance to a pie which has been sliced.


A pie chart is a tool that helps you visualized the relative importance of several diagrams... A histogram is a diagram which graphically depicts the variability in a process

These are best used with categorical data to help us see what percentage of the whole each category constitutes. These Pie charts require all categories to be included in a graph. Every graph always represents their entire segments.

Draw a pie chart to display the information.

Solution:

Total weekly expenditure in house = 250 + 300 + $85

= $635

Find the percentage of total expenditure of each item

Percentage:

Fruits = (250 / 600)100% = 41.6%

Vegetable = (300 / 600) 100 = 50%

Oils = (85 / 600) 100 = 14.1%

If we draw the pie chart, divide the circle into hundred parts. Allocate the percentage parts require for each item.

Uses:

A pie diagram can be used in various applications. For instance, this is mostly used in government to represent cities and the statistical information that relates to income, age, gender and race. A pie chart makes the information more easily and understood as a graphical representation of the statistics.

Hope you like the above example of Pie Chart.Please leave your comments, if you have any doubts.

Binomial Distribution

Binomial Distribution:

Introduction to Binomial Distribution:

The general meaning of a Binomial Distribution is if two mutually exclusive possible outcomes are available means binomial distribution happen.Discrete probability distribution concept is used..It sequence of success in numbers.In binomial distribution,two outcomes are referred.One outcome is success and another one is failure.Probability f success in trials with probability of success on single trial is determined by binomial distribution.p is denote the binomial distribution.


Binomial Distribution is a statistical experiment which means the number of successes in n repeated trials of a binomial experiment. It is also called as Bernoulli distribution or Bernoulli trial.To understand any concept an example is required,now let us look at an example of Binomial Distribution.

For example:

For a clinical trial, a patient may live or die. Here the researcher faces the number of survivors and not how much time the patient lives after treatment.


Properties and Formula for binomial distribution:


For example:

For a clinical trial, a patient may live or die. Here the researcher faces the number of survivors and not how much time the patient lives after treatment.

We take a coin and flipped two times. Here we calculate the count of number of heads(successes). So the binomial distribution is

Number of heads Probability

No head 0.25

One head 0.5

Two head 0.25


Properties of Binomial Distribution:

1. The experiment has n repeated trials.
2. Each trial can have two possible outcomes. One is success and another one is failure.
3. Here the trials are independent.
4. Mean = n * P.
5. Variance = n * P * (1 – P).
6. Standard Deviation = sqrt[ n * P * ( 1 – P ) ].

Binomial distribution Formula:

b(x; n, P) = nCx * Px * (1 - P)n – x

Here the Notation are,

B(x; n, P) = Binomial Probability.

X = successes

N = number of trials

P = Probability of success

nCx = Number of combinations of n trials, x is success.


Example Problem(the binomial distribution):

A die is tossed 6 times. What is the Probability of getting exactly 2 fours?

Solution

Here n = 6, x = 2, probability of success on a single trial = 1/ 6 or 01.167.

Therefore, The binomial probability is,

b( 2; 6, 0.167 ) = 6C2 * ( 0.167 )2 * ( 1 – 0.167)6 – 2

= ( 6! / 2! * (6-2)!) * 0.0279 * ( 0.833)4

= (6! / 2! * 4!) * 0.0279 * 0.481

= 15 * 0.0279 * 0.481

b( 2; 6, 0.167 ) = 0.201. Answer.


Hope you like the above example of Binomial Distribution.Please leave your comments, if you have any doubts.

Poisson Distribution definition:

Poisson distribution definition:
The Poisson distribution theory is explained in detail below:
In statistics,the cases of probability theory consists of the Poisson distribution of large numbers are discrete probability distribution that relates the probability of a many cases done in a fixed time period. When the event occurs with an average rate & independently of the time.Also be used for the some events in other specified intervals such as distance,volume or area .

Formula for Poisson Distribution:

If the expected case of the number of cases in this interval is λ, that the probability there are exactly n occurrences(n being a non-negative integer, n = 0, 1, 2 ...)is equal to

f (n:ג)=(גn* e-ג)/n!

where,

* e is the base of natural logarithm(e = 2.71828...).
* n is the some occurrences of an event happens mutually the probability of which is given by the function in study poison distribution.
* n! is the factorial of n.
* λ is the positive real number is equal to the expected number of occurrences that may occur during the given interval.If the events occurs on average four times per minute, and are interested in the number of events occurring in a ten minutes interval, would use as the model a Poisson distribution with λ = 10×4 = 40.

The study of Poisson distribution can applying to various systems with a large number of possible events, each of rare. A classic example is the nuclear decay of atoms.

Algorithm for poisson distribution

Algorithm:

algorithm :Poisson random number:

init:

Let L ← e−λ, k ← 0 and p ← 1.

do:

Generate uniform random number u in [0,1] and let p ← p × u.

while p > L.

return k − 1.

Hope you like the above example of Poisson Distribution.Please leave your comments, if you have any doubts.

Permutations and combinations:

Permutations and Combinations:
When we talk about the concept of Permutation and Combination,the basic meaning of permutation is rearranging in an ordered fashion,and the combination means the selection of a number of things taking some or all of them at a time.The permutation and combination takes place on different types of objects. The permutation of a different object is the number of different ways they can be ordered i.e. which is first, second, third, etc. If you desire to choose some objects from a larger number of objects, the way you place the chosen objects is also important. When comes to combination, on the other hand, one does not consider the order in which objects were selected or placed, just which objects were selected.

Permutation:

Permutation has two types:

* Permutation with Repetition.
* Permutation without Repetition.

Permutation with Repetition:

When we have n different objects then we have n choices each time. And if we are in a position to choose object r from n objects, the permutations are

n * n * n…..(r times) = n r

P (n, r) = n r

Permutation without Repetition:

In permutation without Repetition, we have to reduce the number of available choices each time. When we have n different objects then we have to reduce 1 from the previous term for each time.

This is like n * (n-1) * (n-2)….

And if we are in a position to choose r objects from n objects, the combination is

P (n, r) = [(n!)/((n-r)!)]

Example:

In how many ways a man can put 4 balls in 3 bags.

Solution:

First ball can put in 3 ways.

Second ball can put in 3 ways.

Third ball can put in 3 ways.

Fourth ball can put in 3 ways.

So the 4 ball can put in 3 * 3* 3* 3 =34 = 81 ways.

Combination:

Combination has two types:

* Combination with Repetition.
* Combination without Repetition.

Combination with Repetition:

We need to do is alter our permutations formula to reduce it by how many ways the objects could be in order but the order is not important here.

C (n, r) = [(n!)/(r!(n-r)!)]

Combination without Repetition:

When we have n different objects and to select r objects with repetition we have a formula

C (n, r) = [(n+r-1)/(r!(n-r)!)]


Example:

Write all the combination of four balls taken one at a time.

Solution:

Here n=4 and r=1

4C1= [(4!)/(1!(4-1)!)]

4C1= [(4 * 3 * 2 *1)/(1 * 3!)]

4C1= [(4*3*2*1)/(1*3*2*1)]

4C1 = 4.

Hope you like the above example of Permutations and Combinations.Please leave your comments, if you have any doubts.

Parabola

Parabola:




Parabola:

Let us learn about the meaning of a Parabola,and also let us learn to solve the equation of a Parabola.

The locus of a point whose distance from a fixed point is equal to its distance from a fixed line is called a parabola. That is a parabola is a conic whose eccentricity is 1.

Standard equation of a parabola :


Given :
Fixed point (F)
Fixed line (l)
Eccentricity (e = 1)
Moving point P(x, y)
Construction :
Plot the fixed point F and draw the fixed line ‘l’.
Drop a perpendicular (FZ) from F to l.
Take FZ = 2a and treat it as x-axis.
Draw a perpendicular bisector to FZ and treat it as y-axis.
Let V(0, 0) be the origin.
Drop a perpendicular (PM) from P to l.
The known points are F(a, 0), Z(− a, 0) and hence M is (− a, y).
By the definition of a conic,
FP / PM = e = 1 ⇒ FP2 = PM2
(x − a)2 + (y − 0)2 = (x + a)2 + (y − y)2
x2 − 2ax + a2 + y2 = x2 + 2ax + a2 which simplifies to y2 = 4ax.
This is the standard equation of the parabola.


Ellipse:

An ellipse is a conic obtained on slicing across obliquely one nappe of a cone.If P is any point on the ellipse and
F1and F2 its foci, the angle subtended by F1P and F2P with the tangent at P are equal and if a source of light or sound is
placed at one focus of an ellipsoidal reflector (surface generated by revolving an ellipse about its major axis) all the
waves will be reflected so as to pass through the other focus



Hyperbola:

A hyperbola is a conic obtained on slicing a double napped cone by a plane parallel to the axis of the cone. The lines from the foci to any point of a hyperbola make equal angles with the tangent at that point. Hence if the surface of a reflector is generated by revolving a hyperbola about its transverse axis, all rays of light converging on one focus are reflected to the other.

Hope you like the above example of Parabola.Please leave your comments, if you have any doubts.

Conditional Probability:

Conditional Probability:

What do we understand by the term Conditional Probability??? And what is the meaning of the term Conditional Probability??? Let us now find answers to these frequently asked questions.Suppose the two events are not independent, that is the occurrence of one depends on the occurrence of other, then how do we compute This can be explained by conditional probability.

Baye's theorem is named after the British mathematician Thomas Bayes who published it in a research paper in 1763. It gives one of the important applications of the conditional probabilities by using the additional information supplied by the experiment or the past records.

Conditional Probability

Let A and B be any two events associated with a random experiment. The probability of occurrence of event A when the event B has already occurred is called the conditional probability of A when B is given and is denoted as P(A/B).

Solving Coin problems

In algebra we deal with the word problems involve pennies, nickels , dimes and half quarters. To solve such type of problems we need prepare tables.

Probability using Z score

By using Z-scores, normal distribution is standardized. We can find probability for standard normal distribution using Z-scores. Using random variable, mean and standard deviation, Z-scores are calculated.

Hope you like the above example of Conditional Probability.Please leave your comments, if you have any doubts.

Properties of Proportions

Learn us study properties of proportions,
The four properties that follow are not difficult to justify algebraically, but the details will not be presented here.
Property 1 (Means-Extremes Property, or Cross-Products Property): If a/b = c/d, then ad =bc. Conversely, if ad = bc ≠ 0, then and .


Example 1: Find a if a/12 = 3/4.
By Property 1:

Example 2: Is 3 : 4 = 7 : 8 a proportion?
No. If this were a proportion, Property 1 would produce


Property 2 (Means or Extremes Switching Property): If a/ b = c/ d and is a proportion, then both d/ b = c/ a and a/ c = b/ d are proportions.
Example 3: 8/10 = 4/5 is a proportion. Property 2 says that if you were to switch the 8 and 5 or switch the 4 and 10, then the new statement is still a proportion.
If 8/10 = 4/5, then 5/10 = 4/8, or if 8/10 = 4/5, then 8/4 = 10/5.
Hope the above explanation helped you.

Orthocentre

Orthocentre:
Let us firstly learn about the definition of Orthocentre.
Definition of Orthocentre of Triangle:

The point of concurrency of the three altitudes of a triangle is called its "Orthocentre". It is generally abbreviated as 'O'.

To locate orthocentre it is sufficient to draw altitudes of any two sides of a triangle. The third altitude will then automatically pass through it.


Theorem on Orthocentre of Triangle:

The Theorem of Orthocentre of Triangle says that in a triangle, the three altitudes pass through the same point.

Given:

In DABC, AD, BE and CF are the altitudes.
To prove Theorem on Orthocentre of Triangle:

AD, BE and CF are concurrent (or pass through the same point)
Construction:

Through A, B and C draw lines parallel to BC, AB and AC respectively. Let these lines meet at P, Q and R forming DPQR.
Proof:

QA || BC and QC || AB by construction.
ABCQ is a parallelogram.

AQ = BC . . . (1)
Similarly BCAR is a parallelogram.

AR = BC . . . (2)
From (1) and (2),

AQ = AR . . . (3)

AD is the perpendicular bisector of RQ.
Similarly we can prove that BE is the perpendicular bisector of PR and CF is the perpendicular bisector of PQ. Thus AD, BE and CF are the perpendicular bisectors of the sides of DPQR. Hence AD, BE and CF pass through the same point. (by theorem 2)

Hope you like the above example of Theorem on Orthocentre of Triangle.Please leave your comments, if you have any doubts.

Properties of whole numbers

Properties of whole numbers:



Whole Numbers:While solving problems we usually come across questions like what do we mean by whole numbers,how can we identify whole numbers.Below is the explanation for these type of frequently asked questions

Meaning of whole numbers:

The set of whole numbers is the set of natural numbers along with zero. so W = the set of whole numbers = 0,1,2,3,............

so Zero is the least number of the set of Whole numbers.

As the whole numbers is an infinite set we cant determine the highest number of this set.

The set of Whole numbers is a subset of Rational numbers.
Let us now learn about the Properties of whole numbers
Properties of Whole numbers:

t is advisable to use a number line inorder to understand the properties of whole numbers.

1. Number 3 < 5 and the number 3 is to the left of the number 5.Hence of an two numbers on a number line, the smaller number is to the left of the greater number.So too,of any two numbers on a number line the number to the left of the other number, is the smaller of the two.

2. There is no whole number to the left of zero on the number line.So zero is the smaller number than each of the numbers to its right on the number line.That means 0 is the smallest or least of the whole numbers.

3. A whole number which is greater than a given whole number by 1 is said to be a successive whole number. 1 is the successive whole number to 0.Every whole number has one successor.

4. There is no whole number left to zero.hence 0 is not a successive whole number of any whole.

Hope you like the above example of Whole Numbers.Please leave your comments, if you have any doubts.

Circumference Of A Circle

Circumference Of A Circle:

Circumference of a Circle:
When we study about a circle the most common question that we come across is what is the circumference of a circle.The answer to this question is very simple,the distance around the circle is Circumference. In a circle, the distance from center to any point of a circle is called radius. And the line which touches two points of the circle and passes through the center is known as Diameter of a circle.

The formula to find the Circumference of a circle C is C = 2 × Pi × r

Which means Circumference of a circle is 2 times the value of pi times r

Where, r = radius of a circle and

Pi=3.142 which is a constant value.

The formula to find the Circumference of a Circle with Diameter is

C=Pi*d

Which means the circumference of a circle is pi times the diameter.

where, d=Diameter of a circle and

Pi=3.142 which is a constant value.

As we know Diameter is equal to radius divided by 2, we can use the first formula when we have the Diameter in a given problem.

D = r / /2

Which means diameter d is radius r divided by 2.

Now, we can write the equivalent formula for the diameter which is radius is 2 times the diameter.

r=2 × D

The circumference of circle units can be measured in linear units, like inches, centimeter etc,


Example of Cicumference of the Circle:


Consider the wheel of a cycle and whose one of the length of the sting is 100cm. Since, in a cycle all the stings are passing thorough the center of cycle’s wheel, all the stings are equal to the diameter. So, now we have the diameter of the wheel. With this we can find the circumference of a wheel with the formula circumference is equal to pi times the diameter.

So the circumference of a cycle’s wheel is, C=pi*D

Here we have diameter d=100cm and pi=3.142

Now C = 3.142 × 100cm

C = 314.2cm

The Circumference of cycle’s wheel whose one of the sting’s length 100cm is 314.2cm.

Hope you like the above example of Circumference Of A Circle.Please leave your comments, if you have any doubts.

Isoceles Triangle

Isoceles Triangle:

Introduction to triangles:Let me first brief you up about Triangles in general,a triangle is a three sided figure,and an isosceles triangle has two congruent faces called legs and a third side called the base. The vertex angle is the angle included by the legs. The other two angles are called base angles. The base angles are congruent.
Isosceles Triangle:

The easiest way to learn about the isosceles triangle is to first identify its properties.

• The unequal face of an isoceles triangle is usually referred to as the 'base' of the triangle.
• The base angles (congruent) of an isosceles triangle are always equal. In the figure above, the angles
• When the 3rd angle is a right angle, it is called a "right isoceles triangle".
• The altitude is a perpendicular length from the base to the topmost vertex.

All isoceles triangles:
- Have angles that add up to 180 degrees
- Have two equivalent sides. The irregular side is called the base.
- Have alike base angles.
- Have areas and perimeters that can be establish using the formulas Area=1/2 X (base X height) and Perimeter=side+side+side
A shape triangle with a right angle is called a right isoceles triangle. Also, all equilateral triangles are isoceles triangles, but not all isoceles triangles are right triangles.

Hope you like the above example of Isoceles Triangle.Please leave your comments, if you have any doubts.

Theorem

Theorem:

Introduction:

A Statement that requires a proof is called a theorem.
A theorem is a generalised statement, which can be proved logically. A theorem has two parts, a hypothesis, which states the given facts and a conclusion which states the property to be proved. The two statements given above are examples of theorems.
Theorems are proved using undefined terms, definitions, postulates and occasionally some axioms from algebra.
A theorem is a generalised statement because it is always true. For example the statement or the proposition “If two straight lines intersect, then the vertically opposite angles are equal” is true for any two straight lines intersecting at a point. Such a statement is called the general enunciation.
We can understand Theorems better with the help of an example:
As we can see the above figure,which says that,

Congruent angles are the one that have have the same angle measures. The theorem based on congruent angles is mentioned below

Statement

If two lines intersect, then the vertically opposite angles are congruent.

To Prove

Proof:

Similarly it can be proved that

Hence the theorem is proved.

Hope you like the above example of Theorems.Please leave your comments, if you have any doubts.

Arithmetic progression:

Arithmetic progression:

There are two types of progressions .They are arithmetic and geometric progression. An Arithmetic progression which consists of the sequence of numbers and the terms except the first can be obtained by adding one number to its preceding number. Arithmetic progression is denoted as two consecutive numbers arrangement of progression which is constant.
Let us now learn about the use of arithmetic progression,

• An arithmetic succession is a sequence of numbers such that the difference of some two successive members of the series is a constant.

• Arithmetic progression is used to work out the calculation of the succession and the product of the sequences in the problem.
• S_n= (a+l) (or) {2a+ (n-1) d} Here,
  a-first term of the sequence
  n -total number of sequence
  d - Difference between any two numbers in series
  Here l is the last term
• Let us also learn about properties of arithmetic progression:

• When we add or subtract any constant number with all the terms of the sequence, the arithmetic sequence remains an arithmetic sequence.

Example:
5, 7, 9, 11, 13, 15, 17….. is an A.P with common difference 2.
Add 3 with all the terms,
8, 10, 12, 14, 16, 18, 20…. Is also an A.P with common difference 2.

• When we multiply or divide by a non-zero constant with all the terms of the sequence, the arithmetic progression sequence remains an arithmetic progression.

• Hope you like the above example of Arithmetic progression:.Please leave your comments, if you have any doubts.

Congruent Triangles

We will learn about Congruent triangles,
Three noncolinear points determine a triangle. Draw the three segments connecting the three pairs of points and find three sides, and 3 interior angles, hence the name triangle.

Two triangles are congruent if one can be moved on top of the other, so that edges and vertices coincide. The corresponding sides have the same lengths, and corresponding angles are congruent.


Assume the edges of one triangle are the same lengths as the corresponding edges of another triangle. Move the first triangle onto the second so that the bases coincide. Do the apexes also coincide? Both apexes are x units away from the left end of the base, and y units from the right end of the base. Since the triangles are oriented the same way, both apexes lie above the base. Draw a circle of radius x centered at the left end of the base, and a circle of radius y centered at the right end of the base. These circles intersect in precisely two points, one above the base and one below. Thus there is only one possible location for the apex. Both apexes coincide, and the first triangle lies directly on top of the second. Corresponding angles coincide, and are congruent. This method of proving triangle congruence is called SSS, for side side side.

Hope the above explanation helped you, now let us study about types of triangles.