Category: Maths Notes PPT

Sample Size And Sampling Techniques In Research

Before proceeding with the sampling techniques/methods, let’s first understand what exactly sampling is. Sampling is a technique of sorting out individual representatives or a division of the population to extract statistical derivation from them as well gauge features of the entire population. For the same, there are different sampling methods. That said, the Sampling methods refer to the technique of selecting members from the population to involve in the study by using different methods.         

                                 

Types Of Sampling Methods

Sampling in statistical study is of two types i.e. – probability sampling and non-probability sampling. Let’s closely review the two methods of sampling that can be implemented in any market survey.

1. Probability Sampling

Probability sampling is a sampling technique where a researcher sets the benchmark of selection based on a few criteria and selects members of a large population randomly. All the members bear an equal opportunity to be participating in the sample with this selection criterion. It is a conclusive type of sampling.

For example, in a community of 10,000 residents, every member will have a 1/10,000 chance of being chosen to be a part of a sample. Probability sampling is free of prejudice. Thus, removes bias in the population and provides all members a fair opportunity to get involved in the sample based on a fixed process.

Now, probability sampling also has 4 different methods under its kingdom which are as below

                     

Types Of Probability Sampling With Examples:

1. Simple Random Sampling: One of the choicest probability sampling methods that saves time and resources. It is a well-grounded technique of collecting information where every single member of a population is selected randomly, solely unintentionally. Each individual will have the same probability of being opted for to be a part of a sample. For example, in a society of 1000 residents, if the facilities head decides on conducting sanitization activities, it is largely possible that they would prefer picking chits out of a box. In such a case, each of the 1000 residents has an equal chance of being chosen.

2. Systematic Sampling: Survey creators use the systematic sampling method to select the sample members of a population at regular or systematic intervals. It needs the choice of an initial point for the sample and sample size that can be repeated at systematic intervals. This type of sampling technique has a pre-established range, and thus is the least time-consuming. For example, a researcher seeks to obtain a systematic sample of 1000 people in a population of 20,000. The researcher numbers each component of the population from 1-20000 and will select every 10th individual to be a part of the sample (Total population/ Sample Size = 20000/1000 = 20).

3. Cluster Sampling: Cluster sampling is a technique where the researchers segment the whole population into groups or clusters that exemplify a population. Clusters are determined in a sample on the basis of demographic like age, gender, location, education etc. This makes it quite easier to procure productive derivation from the feedback. For example, if the Indian government intends to assess the number of foreign immigrants living in the country, they can divide it into clusters based on states such as Maharashtra, Karnataka, Tamil Nadu, Andhra Pradesh, West Bengal, Uttar Pradesh, Uttarakhand etc. This way of a survey is more effective as the outcomes will be categorized orderly into states and renders insightful immigration data.

4. Stratified Random Sampling: It is a sampling technique in which the surveyor segments the population into smaller groups that don’t overlap but indicate the whole population. While sampling, these groups can be arranged and then pull out a sample from each group distinctively. For example, a surveyor intends to evaluate the purchasing preferences of people belonging to different annual income groups and create echelon (groups) as per the annual family income. E.g. – less than 5-6lacs–, 10-15lacs, 20-30 lacs etc. This way, the surveyor concludes the attributes of people belonging to different income groups. This will further help Marketers to assess which income groups to target and which ones to eradicate in order to form a blueprint that would yield favorable outcomes. .

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2. Non-probability Sampling

Non-probability sampling is another sampling technique where the researcher selects members for research at random. The technique does not include a fixed or predefined selection process. This makes it quite challenging for all components of a population to have equal chances to be part of a sample. It is an exploratory type of sampling.

Types Of Non-probability Sampling With Examples

There are 4 types of non-probability sampling methods of which each sampling method has a specific purpose. Let’s look closely into their functionality

1. Convenience Sampling: This technique relies on the ease of access such as to survey buyers at a mall or passers-by on a busy street by getting in touch with the subjects. The elements of the sample are solely selected based on accessibility and not characteristically. Thus it makes for a practical method of sampling when there are time and cost restraints. For example, travel agencies generally conduct convenience sampling at a mall or public places to distribute upcoming events– they do that by handing out leaflets randomly.

2. Judgmental or Purposive Sampling: A judgmental sample is drawn based on the judgment of the researcher. What Researchers take into consideration is solely—the intent of the study, together with the understanding of the target audience. For example, when researchers seek to gain insight into the thought process of people willing to work in foreign countries. The parameter is: “Are you willing to work abroad…?” and those who respond with a “YES” are included in the sample.

3. Quota Sampling: The selection of members happens to be based on a predefined standard. In such a case, a sample is created resting upon a particular characteristic. The sample formed will have the similar attribute as that of found in the total population.

4. Snowball Sampling: The sampling technique is usually when the subjects are difficult to locate. Sampling method is also widely used in events where the concern is extremely sensitive and not openly discussed—says for example to collect data about HIV Aids+. For example, it will be exceptionally difficult to trace illegal immigrants. In this case, employing the snowball theory can help researchers track some divisions to interrogate and extract results.

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Practice Problem 

Pr
actice Problem 1

Each student in a college has a roll number. Administrator has a computer generate 202020 random identification numbers and those students are asked to participate in a survey.

What Type Of Sampling Method Is This?

Answer: Stratified random sampling

Practice Problem 2

Example A local hotel chain intends to survey its visitors one day, so they randomly sent out questionnaires that day and surveyed every visitor on the venue.

What Type of Sampling Will Be Best?

A cluster sample as it will have every member in access. Moreover it’s good when each individual reflects the population as a whole.

Fun Facts

• The main advantage of probability sampling methods is that they ensure that the sample selected is indicative of the population

• There are wide applications of probability sampling.

• The likeliness to prejudice in sample obtained is negligible to non-existent

• Probability sampling take into account vast and diverse population and the data is not fudged towards one demographic

• Random sampling is considered to be a wholesome form of probability survey sampling

• Systematic sampling is often used instead of random sampling

• Due to minimal error, Stratified sampling is frequently incorporated probability method than random sampling

• Non-probability sampling, in most situations, leads to skewed outcomes

• A non-probable sample is much more useful for fundamental stages of research

• Quota sampling is the quickest method of obtaining samples.

In our day to day life, we come across various objects having different shapes and sizes which are based on parameters like physical properties such as length, breadth, diameter, etc and sometimes it also depends on the material. But no matter how different their dimensions are, all the objects are matter and occupies space. These objects are also referred to as three-dimensional or solid shapes which can be viewed from different sections. Visualization of solid shapes helps us to understand the solid object.  

What are solid Shapes?

Solid shapes are the objects having three-dimensional shapes such that the position of any point can be explained by using three coordinate axes known as x-axis, y-axis, and z-axis.  Many objects that we see in your day to day life as a bed, cylinder, cupboard, etc are three-dimensional objects occupying some shape and having length, breadth, height, and depth.

Properties Of 3-D Shapes

There are four properties that set three-dimensional shape apart from two-dimensional shapes and these properties are faces, vertices, edges, and volume. These properties not only allow you to determine whether the shape is 2D or 3D but it also helps you to understand which type or division of solids it belongs to. 

Faces, Edges and Vertices

A face is a two-dimensional surface as one of the surfaces of a three-dimensional solid. An edge is the meeting line of two faces just like how sky and land appear to meet at the horizon. Vertex is the point or tip of the corner of three-dimensional geometric shapes. Thus, a solid figure has faces (sides/ surfaces having areas), edges (the meeting line of two surfaces) and vertices (corners/ tips).

Faces

A face is a flat or curved surface of a solid shape. For example, a cube has six faces whereas a cylinder has three faces and a sphere has only one face.

Edges

An edge is where two faces meet it appears to be a straight line. For example, a cube has 12 edges, a cylinder has two edges and a sphere has no edges at all.

Vertices

A vertex is a corner of the solid where the edges meet. A lot of vertexes together is known as vertices. For example, a cube consists of eight vertices, a cone consists of one vertex and a sphere has no vertex.

Cross Sections Of Solid Shapes

When we cut a solid object, we get a surface which is called cross-section and it has an area too. In other words, a cross-section is a shape we get when after cutting an object straight through. It is more like a view into the inside of the object by cutting through it. A cross-section is the intersection of a three-dimensional figure with a plane that is more like a face you obtain by slicing through a solid object. A cross-section is always two-dimensional and the area of the face of the cross-section depends on the orientation (angle) of the plane while cutting the object. Cross-sections are usually either parallel or perpendicular to the base but it can be in any direction.

Difference Between Section And Cross Section

The main difference between a Section and a Cross Section is that a section is the cutting of a solid by a plane, whereas a cross-section is actually the surface or the face having an area which is exposed when we cut the object. A section refers to a closeup of a particular section or part of the design that can be any angle but a cross-section refers to a view of something that has been cut across to show the interior of the object.

Ways to View The Sections of solids

There are three ways to view the section of a Solid Shape:

  •  Viewing the cross-sections

  •  Using shadows

  •  Viewing at certain angles

A solid can be viewed from different angles such as from the front, side and top. On the other hand, cutting or slicing a solid will show the cross-section of the object. Observing the two-dimensional shadow of a three-dimensional solid is also another way of viewing a solid. Shadows of three-dimensional solids are of different sizes depending on its position and the source of light.

1. Cutting or Slicing

We have already read about the cross-section of solids which is basically the exposed surface of a solid that you get when you make a cut through it. The original face cannot be retained once the object is cut therefore the cross-section is a surface “inside” the object.

To view the cross-section of 3D objects you can cut or slice the object from any place at any angle. You can cut an object horizontally, vertically or from any angle. 

2. Shadow Play

You can view the cross-section of a solid by using shadow which requires a source of light a, for example, a torch or sun or bulb, etc. You can view solids such as cuboid, cone, sphere, etc by keeping them in front of a screen and bring the torch in front of the solid the opposite side of the screen. You can view the shadow of the object on the screen. The size of the shadow depends on the angle and distance of the light source and the screen from each other and the object.

Two or more figures may have the same shape but not necessarily same size then those figures are said to be similar.

However, if two or more figures have the same shape and same size then those figures are said to be congruent.

Note:

1. The congruent figures are similar but, the converse is not true, i.e. similar figures need not be congruent.

2. All regular polygons of the same number of sides are similar like, all equilateral triangles, all squares, etc. are similar.

3. All circles are always similar irrespective of their diameter or radius.

4. The similarity of figures is denoted by the ‘~’ symbol.

Similarity of Two Polygons

Two polygons of the same number of sides are similar, if 

1. their corresponding angles are equal and 

2. their corresponding sides are in the same ratio (or proportion).

Similarity of Two Triangles

Since, we know that triangle is the smallest three-sided polygon. So, we can state the same conditions for the similarity of two triangles. That is: Two triangles are similar, if 

1. their corresponding angles are equal and 

2. their corresponding sides are in the same ratio (or proportion).

In geometry, correspondence means that a particular part of one polygon relates exactly to a similarly positioned part of another polygon. Even if two triangles are oriented differently from each other, if we rotate them to orient in the same way and see that their angles are alike, we can say those angles are corresponding angles of the two triangles.

Example:

In the given figure, two triangles ΔABC and ΔPQR are similar only if,

i)       [angle A{text{ }} = {text{ }}angle P,{text{ }}angle B{text{ }} = {text{ }}angle Q{text{ }}and{text{ }}angle C{text{ }} = {text{ }}angle R]

 ii)       [frac{{AB}}{{PQ}} = frac{{BC}}{{QR}} = frac{{AC}}{{PR}}]

Hence if the above-mentioned conditions are satisfied then we can say that ΔABC ~ ΔPQR

Note that if corresponding angles of two triangles are equal, then they are known as equiangular triangles. 

A famous Greek mathematician Thales gave an important truth relating to two equiangular triangles which is as follows: The ratio of any two corresponding sides in two equiangular triangles is always the same. It is believed that he had used a result called the Basic Proportionality Theorem (now known as the Thales Theorem) for the same.

Basic Proportionality Theorem (or Thales Theorem) 

Statement: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.

Given: A triangle ABC in which a line parallel to side BC intersects other two sides AB and AC at D and E respectively.

To prove:  [frac{{AD}}{{DB}} = frac{{AE}}{{EC}}]

Construction: Let us join BE and CD and then draw DM ⊥ AC and EN ⊥ AB.

Proof:   Since, area of Δ ADE = ([frac{1}{2}]× base × height) = [frac{1}{2}] × AD × EN.

Area of Δ ADE is denoted as ar(ADE).

So, ar(ADE) = [frac{1}{2}]× AD × EN

Similarly, ar(BDE) = [frac{1}{2}] × DB × EN,

                 ar(ADE) = [frac{1}{2}] × AE × DM and ar(DEC) = [frac{1}{2}]× EC × DM.

Therefore, [frac{{arleft( {ADE} right)}}{{arleft( {BDE} right)}} = frac{{frac{1}{2} times AD times EN}}{{frac{1}{2} times DB times EN}} = frac{{AD}}{{DB}}]                                         (1) 

                                                                                                                  

And           [frac{{arleft( {ADE} right)}}{{arleft( {DEC} right)}} = frac{{frac{1}{2} times AE times DM}}{{frac{1}{2} times EC times DM}} = frac{{AE}}{{EC}}]                                        (2)

 Since, Δ BDE and DEC are on the same base DE and between the same parallels BC and DE. 

So, ar(BDE) = ar(DEC)                                                                   (3)

Therefore, from (1), (2) and (3), 

we obtain:   [frac{{AD}}{{DB}} = frac{{AE}}{{EC}}]           (Proved)

Similarity Criterion of a Triangle

There are three similarity criteria of triangles which are used to solve the problems based on similar triangles. They are as following:

1. AA (or AAA) or Angle-Angle similarity criterion

2. SSS or Side-Side-Side similarity criterion

3. SAS or Side-Angle-Side similarity criterion

AA (or AAA) or Angle-Angle similarity criterion

Statement: If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar.

Also, if two angles of a triangle are respectively equal to two angles of another triangle, then by the angle sum property of a triangle their third angles will also be equal. 

Therefore, AAA similarity criterion can also be stated as follows:

If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar. This may be referred to as the AA similarity criterion for two triangles.

SSS or Side-Side-Side Similarity Criterion

Statement: If in two triangles, sides of one triangle are proportional to (or in the same ratio of) the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar.

SAS or Side-Angle-Side Similarity Criterion

Statement: If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar.

Areas of Similar Triangles

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. i.e.,

For given two triangles ABC and PQR such that Δ ABC ~ Δ PQR     

Then, [frac{{arleft( {ABC} right)}}{{arleft( {PQR} right)}} = {left( {frac{{AB}}{{PQ}}} right)^2} = {left( {frac{{BC}}{{QR}}} right)^2} = {left( {frac{{AC}}{{PR}}} right)^2}]

Solved Examples:

Q.1. Give Examples of Pairs of (i) Similar Figures (ii) Non-Similar Figures.

Ans. (i) similar figures: all equilateral triangles, all circles.

         (ii) non-similar figures: a circle and a square, a triangle and a trapezium.

Q.2. In the Figure Given Below, DE || BC. Find EC.

Ans. Let EC = x cm.

        It is given that DE || BC.

By using Basic proportionality theorem, we obtain

   [frac{{AD}}{{DB}} = frac{{AE}}{{EC}}]

⇒ [frac{{1.5}}{3} = frac{1}{x}]

⇒ [X = frac{3}{{1.5}}]

⇒ x = 2

So, EC = 2 cm.

Q.3. Let Δ ABC ~ Δ DEF and their Areas be 64 cm2 and 121 cm2 Respectively. If EF = 15.4 cm then Find BC.

Ans.  It is given that Δ ABC ~ Δ DEF

So, [frac{{arleft( {ABC} right)}}{{arleft( {DEF} right)}} = {left( {frac{{AB}}{{DE}}} right)^2} = {left( {frac{{BC}}{{EF}}} right)^2} = {left( {frac{{AC}}{{DF}}} right)^2}]

Given that, 

EF = 15.4 cm

ar(ABC) = 64 cm2

ar(DEF) = 121 cm2

So, [frac{{arleft( {ABC} right)}}{{arleft( {DEF} right)}}]=[{left( {frac{{BC}}{{EF}}} right)^2}]

 ⇒ [left( {frac{{64{text{ }}cm2}}{{121{text{ }}cm2}}} right) = frac{{BC2}}{{15.4{text{ }}cm}}]

⇒ [frac{{BC}}{{15.4{text{ }}cm}} = frac{8}{{11}}{text{ }}cm]

⇒ [BC = frac{{8 times 15.4}}{{11}}]= 11.2 cm

Sine, cosine, and tangent (abbreviated as sin, cos, and tan) are three primary trigonometric functions, which relate an angle of a right-angled triangle to the ratios of two sides length. The reciprocals of sine, cosine, and tangent are the secant, the cosecant, and the cotangent respectively. Each of the six trigonometric functions has corresponding inverse functions (also known as inverse trigonometric functions). The trigonometric functions also known as the circular functions, angle functions, or goniometric functions are widely used in all fields of science that are related to Geometry such as navigation, celestial mechanics, solid mechanics, etc.

Read below to know what is a sine function, cosine function, and tangent function in detail.

Sine Cosine Tangent Definition

A right-angled triangle includes one angle of 90 degrees and two acute angles. Each acute angle of a right-angled triangle retains the property of the sine cosine tangent. The sine, cosine, and tangent of an acute angle of a right-angled triangle are defined as the ratio of two of three sides of the right-angled triangle.

As we know, sine, cosine, and tangent are based on the right-angled triangle, it would be beneficial to give names to each of the triangles to avoid confusion.

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• “Hypotenuse side” is the longest side.

• “Adjacent side” is the side next to angle θ.

• “Opposite side” is the side opposite to angle θ.

Accordingly,

Sin θ = Opposite side/Hypotenuse

Cos θ = Adjacent/Hypotenuse

Tan θ = Opposite/Adjacent

What is the Sine Function?

In the right triangle, the sine function is defined as the ratio of the length of the opposite side to that of the hypotenuse side.

Sin θ = Opposite Side/ Hypotenuse Side.

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For example, the sine function of a triangle ABC with an angle θ  is expressed as:

Sin θ = a/c

What is the Cosine Function?

In the right triangle, the cosine function is defined as the ratio of the length of the adjacent side to that of the hypotenuse side.

Cos θ = Adjacent Side/Hypotenuse Side

Example:

Considering the figure given above, the cosine function of a triangle ABC with an angle θ is expressed as:

Cos θ = b/c

What is the Tangent Function?

In the right triangle, the tangent function is defined as the ratio of the length of the opposite side to that of the adjacent side.

Tan θ = Opposite Side/Adjacent Side

Example:

Considering the figure given above, the cosine function of a triangle ABC with an angle θ is expressed as:

Tan θ = a/b

Sine Cosine Tangent Table

The values of trigonometric ratios like sine, cosine, and tangent for some standard angles such as 0°, 30°, 45°, 60°, and 90° can be easily determined with the help of the sine cosine tangent table given below. These values are very important to solve trigonometric problems. Hence, it is important to learn the values of trigonometric ratios of these standard angles.

The sine, cosine, and tangent table given below includes the values of standard angles like 0°, 30°, 45°, 60°, and 90°.

Sine, Cosine, and Tangent Table

Did You Know?

• Sine and Cosine were introduced by Aryabhatta, whereas the tangent function was introduced by Muhammad Ibn Musa al- Khwarizmi ( 782 CE – 850 CE).

• Sine Cosine and Tangent formulas can be easily learned using SOHCAHTOA. As sine is opposite side over hypotenuse side, cosine is adjacent side over hypotenuse side, and tangent is opposite side over the adjacent side.

Solved Examples:

1. Find Cos θ with respect to the following triangle.

Ans: To find Cos θ, we need both adjacent and hypotenuse side.

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The adjacent side in the above triangle is, BC = 8 Cm

But, the hypotenuse side i.e. AC  is not given.

To find the hypotenuse side, we use the Pythagoras theorem

AC² = AB² + BC²  =  6² + 8² = 100

Hypotenuse side, AC = √100 or 10 cm

Cos θ = Adjacent/Hypotenuse = 8/10

                               = 4/5

Therefore, Cos θ = 4/5

2. Find the value of  Sin 45°, Cos 60°, and Tan 60°.

Solution: Using the trigonometric table above, we have:

Sin 45°  = 1/√2

Cos 60° = 1/2

Tan 45°= 1

What is the Sphere?

A sphere can be regarded as an absolute symmetrical circular shaped object in a three dimensional space. In a three dimensional space, all the points on the surface of the sphere are at the same distance from a fixed point which is regarded as the center of the sphere. The straight line that connects the center of the sphere to any point on its surface is called the radius of the sphere which is generally represented by the letter ‘r’. Diameter of a sphere is that longest line which passes through the center of the sphere and touches its surface at two different points. 

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Important Sphere formula

1. Diameter of a Sphere:

The diameter of a sphere is the straight line that is passing through the center of the sphere and touches two points on either side of its surface. The diameter of a sphere is always two times its radius. If the radius of the sphere is ‘r’, then its diameter is given by the formula:

D = 2 x r

2. Circumference of the Sphere:

Circumference of a sphere can be calculated as 2π times its radius. Circumference of a sphere and that of a circle is given by the same formula:

C = 2 π r

Here, π is a constant and its value is 3.14 or 22/7. So, the circumference of a sphere can also be computed as 6.28 times or 44/7 times its radius. 

3. Total Surface Area of Sphere

The total surface area of a sphere is the same as its curved surface area because the sphere does not have any lateral surfaces. The formula obtained by deriving surface area of a sphere is written Mathematically as:

TSA = 4 π r2

In the above equation, 

TSA is the total surface area of a sphere. It can be simply stated as surface area. 

π is a constant and its value is equal to 3.14 or 22/7

‘r’ represents the value of the radius of the given sphere 

So, the formula of deriving surface area of a sphere is equal to 4π times or 12.56 times or 88/7 times the square of the radius of the sphere. 

4. Volume of a Sphere

Deriving volume of a sphere is the same as finding the total space available within the surface of the sphere. The mathematical formula of deriving volume of a sphere is given as:

V = 4/3 π r3

In the above equation,

‘V’ is the volume of the sphere

π is a constant and its value is equal to 3.14 or 22/7

‘r’ represents the value of the radius of the given sphere 

So, the formula for the deriving volume of a sphere can be stated as 4π/3 times or 4.19 times or 88/21 times the cube of the radius of the sphere whose volume is to be determined.

Worked Examples of Sphere Formula

1. Calculate the diameter and the circumference of a sphere whose radius is 7 cm.

Solution:

Given: Radius of the sphere = 7cm

Diameter of the sphere is calculated as:

D = 2 x r

D = 2 x 7 

D = 14 cm

Circumference of the sphere is found by the formula

C = 2 x π x r

C = 2 x (22/7) x 7

C = 2 x 22 

C = 44 cm

Therefore, the diameter and circumference of the sphere are 14 cm and 44 cm respectively.

2. Find the total surface area and the volume of a sphere whose radius is 14 cm.

Solution:

Given: Radius of the sphere = 14 cm

The formula for the deriving surface area of a sphere is:

A = 4 π r2

A = 4  x (22/7) x (14)2

A = 4 x (22/7) x 14 x 14

A = 4 x 22 x 28

A = 2464 cm2

The volume of a sphere is found using the formula:

V = (4/3) π r3

V = (4/3) x (22/7) x (14)3

V = 11494.04 cc

Therefore, the volume and total surface area of a sphere of radius 14 cm are 11494.04 cc and 2464 cm2 respectively.

3. The volume of a sphere is found to be 729 cc. Find its radius.

Solution:

Given: Volume of the sphere = 729 cc

The formula for deriving volume of a sphere is 

V = 4/3 π r3

729 = (4/3) (22/7) r3

729 = (88/21) r3

r3 = (729 x 21) / 88

r3 = 173.97

r = ∛173.97

r = 5.58 cm

Therefore, the radius of the sphere is 5.58 cm

Fun Facts About What is the Sphere

• By deriving surface area of a sphere formula, it was found by Archimedes that it was the same as the lateral surface area of a cylinder with the base radius equal to that of the sphere and the height equal to the diameter of the sphere. 

• The sphere and circle are not the same. The circle is a two-dimensional closed plane geometric figure whereas a sphere is a three-dimensional circle. 

Have you ever wondered why students are a little uneasy with the subject of Mathematics? Is the subject actually difficult to understand and study? Is it really hard to score good marks in math? Will a student ever fall in love with the subject?

The subject experts at patiently have studied the challenges of students across their network and have made the following conclusions – 

• Students find a subject difficult when they fail to understand the basics clearly

• The teaching method and learning techniques also plays a role in building a student’s relationship with a subject 

• The guidance and the reading materials made available to the students also decide their interest in the subject

Considering all these problems, our team tries their best to make reading more interesting and fun for the students. Students can download the free reading materials where concepts are explained in the easiest language. Video lectures are made available to make students understand better. 

This particular article brings another mathematical concept, explained in detail for the students to grasp the concept and get familiar with mathematical concepts.

Table of content – 

• Square Root – Introduction

• Square Root Definition

• Method of finding the square root

• Prime factorization method

• Solved examples 

• Fun facts 

• Frequently asked questions
  

Square Root Basics

We all are aware of a geometrical shape, the square. Square is a geometrical shape which has four sides of equal length and angles equal to 900. Square, being a two-dimensional shape, covers a specific surface of the plane. This region covered by the square is called its area. Area of a square is calculated as the side x side. If the area of the square is given and its side is to be determined then we use an operation in Mathematics called the square root. For example, if the area of a square is 9 sq. units, then its side measures 3 units which is calculated as the square root of 9.

Square Root Definition

Square of a number is another number obtained by multiplying the number by itself. Square root is the inverse operation of square. Square root of a number is that number which when multiplied by itself, gives the number whose square root is to be determined as the answer. For example, when 7 is multiplied by itself, the product obtained is 49. Therefore, we can say that the square root of 49 is 7. Square root of a number is represented by the symbol ‘√’. It can also be represented exponentially as the number to the power ½ . The square root of a number ‘A’ can be represented as √A or A^1/2. Any number in Mathematics will have two roots of equal magnitude and opposite sign.

Methods to Find Square Root of a Number

Square root of a number can be determined by various methods. A few popular methods used to find the square root of a number are:

1. Guess and Check Method.

2. Average Method.

3. Repeated Subtraction Method.

4. Prime Factorization method.

5. Long Division Method.

6. Number Line Method.

The repeated subtraction method and prime factorization method is applicable only for perfect square numbers. Perfect square numbers are the numbers whose square roots are integers. The examples for perfect square numbers are 1, 4, 9, 16, 25 ……

How to Find the Square Root of a Number by Prime Factorization Method?

Prime factorization method is a method in which the numbers are expressed as a product of their prime factors. The identical prime factors are paired and the product of one element from each pair gives the square root of the number. This method can also be used to find whether a number is a perfect square or not. However, this method cannot be used to find the square root of decimal numbers which are not perfect squares.

Example: Evaluate the root of 576.

Solution: 576 is factorized into its prime factors as follows.

So, 576 can be written as a product of prime numbers as: 576=2×2×2×2×2×2×3×3

Square root of 576 = 2×2×2×3=24

Square Root by Prime Factorization Example Problems

1. Find the square root of 1764 using the prime factorization method.

Solution: Step 1: The given number is resolved into its prime factors.

1764=2×2×3×3×7×7

Step 2:

Identical factors are paired. 

1764=2×2×3×3×7×7

Step 3: One factor from each pair is chosen and the product is found to get the square root. √1764=2×3×7

                    √1764=42

2. Check whether 11025 is a perfect square or not. If it is a perfect square, find its square root by factorization method. 

Solution:

Using prime factorization method, 11025 can be written as the product of its primes as:

11025=3×3×5×5×7×7

All the prime factors can be grouped into pairs of identical factors. No prime factor is left all alone. Hence 11025 is a perfect square number.

√11025=3×5×7=105

 

3. Find the smallest number to be multiplied by 8712 to make it a perfect square number.

Solution:

Using the prime factorization method, 8712 can be factorized as

8712=2×2×2×3×3×11×11

When the identical factors are paired, 8712 can be written as:

8712=2×2×2×3×3×11×11

So, the number 8712 should be multiplied by 2 in order to get a perfect square number.

Fun Facts:

• Any real number has two square roots: a positive root and a negative root. Both the roots are the same in magnitude but the signs are opposite. So, the square root of the number ‘x’ can be written as ±√x.

• The square root of a square of any number is the number itself.

• The square root of non-perfect square numbers cannot be determined using the prime factorization method. However, one can determine the number to be multiplied or divided by the given number to make it a perfect square.