30-Basics of Trigonometry - 17 Important key points

Click here for the previous basics of trigonometry.

So many books solved this problem with so many different methods. I found some easier methods for this problem. so I would like to share this method with you. I am sure, that you may also find some easier methods than this. so Please come forward to share your method also.

Like this, you can apply the technique to make use of simple calculations.

I hope, you will definitely implement this technique to save your valuable time for Examinations or tests. The same logical thinking can be applied to improve your thinking level.

I simply don't want to tell you to go with this simple method of calculation. I want that slowly, you need to mold yourself and improve your thinking level. Only using this technique for solving some problems and getting good scores will not serve my purpose. I want that the entire world needs to improve its thinking skills so that everyone can invent so many beautiful things in human life.

This is possible only with the help of students like you along with your parents.

Please come ahead and start this beautiful work for yourself and show the world that we can also build the world with powerful thinking levels as everyone is competent.

Click here for the next basics of trigonometry.

ANIL SATPUTE

29-Mensuration Problem 4

Once again, it is a problem from Student of 10^th Standard / 10^th grade. I would like to share the method of solving the problem & its presentation with you. Please go through it & make the habit of applying the same technique to solve other problems & please share this technique with your friends & relatives. (Please note one thing that “By distributing our knowledge, it will boost & by keeping it only with us, it will reduce”).

   

The Problem is very simple. Please see the following Problem.

The Cylindrical vessel of radius 15 cm contains water up to the height as 20 cm. A spherical metallic ball is dropped into the water & the water level in the vessel increases by 10.24 cm. Find the radius of the metallic ball. (π = 3.14) (Note: Don’t use a calculator)

Now we will the Simple method of calculations. Please understand the concepts / steps given in box Brackets [ ].

Please see the diagram related to the problem.

Given: 

1) The water level in the cylindrical vessel is increased by 10.24 so height (h) = 10.24 cm

2) Radius of the cylindrical vessel  r₁ = 15 cm

3) The height of water in the cylindrical vessel = 20 cm (This height is not required for our calculations. It is given only to merge the metallic ball completely into the water)

To find: The radius of the Base of a cone.

Solution:

1) According to the problem, 

     The volume of the Spherical Metallic Ball = Volume of the water displaced 

                                                     4/3 π r₂³ =  π r₁² h

                                                      4/3  r₂³  =   r₁² h

                                                             r₂³  =   3 r₁² h / 4

                                                             r₂³  =   3 x 15 x 15 x 10.24 / 4

                                   [Multiply Numerator & denominator by 100]

                                                             r₂³  =   3 x 15 x 15 x 1024 / 4 x 100

[Taking 3 from first 15 and another 3 from second 15 as we want RHS as perfect Cube]

                                                             r₂³  =   3 x 3 x 5  x 3 x 5 x 1024 / 4 x 100

                                   [Re-arranging the terms]

                                                             r₂³  =   (3 x 3 x 3)  x 5 x 5 x 1024 / 4 x 100

[Taking 4 from 1024 as we have 5 x 5 = 25 and when we multiply 25 by 4 we get 100 which will get cancelled with 100 of the denominator]

                                                             r₂³  =   (3 x 3 x 3)  x 5 x 5 x  4 x 256 / 4 x 100

[Taking 100  from Numerator and denominator, we get]

                                                             r₂³  =   (3 x 3 x 3)  x 256 / 4 

[Dividing 256 by 4 from denominator, we get]

                                                             r₂³  =   (3 x 3 x 3)  x 64

                                                             r₂³  =   (3 x 3 x 3)  x (4 x 4 x 4)

                                                             r₂    =   (3 x 4)

                                                             r₂    =   (12)

Therefore the radius of the spherical metallic ball is 12 cm.

Like this, you can apply the technique to make use of simple calculations.

I hope, you will definitely implement this technique to save your valuable time of Examination or test. The same logical thinking can be applied to improve your thinking level.

I simply don't want to tell you to go with this simple method of calculation. I want that slowly, you need to mold yourself and improve the thinking level. Only using this technique for solving some problem and getting good scores will not serve my purpose. I want that the entire world needs to improve the thinking skills so that everyone can invent so many beautiful things in the human life.

This is possible only with the help of students like you along with your parents.

Please come ahead and start this beautiful work for yourself and show to the world that we can also build the world with powerful thinking level as everyone is competent.

I thank Mrs. Jyoti Anil Satpute & Master Abhishek Anil Satpute for creative drawing work.

I also thank Mr. Aaswad Anil Satpute for solving the software issues. 

Anil Satpute

28-Mensuration Problem 3

Once again, it is a problem from Student of 10^th Standard / 10^th grade. I would like to share the method of solving the problem & its presentation with you. Please go through it & make the habit of applying the same technique to solve other problems & please share this technique with your friends & relatives. (Please note one thing that “By distributing our knowledge, it will boost & by keeping it only with us, it will reduce”).

   

The Problem is very simple. Please see the following Problem.

The Volume of a cone of height 5 cm is 753.6 cubic centimeters. Find the radius of the base of the cone. (π = 3.14) (Note: Don’t use a calculator)

Now we will the Simple method of calculations. Please understand the concepts / steps given in box Brackets [ ].

Given: 1) The volume of a cone = 753.6 Cubic Centimeter
           2) The height of a cone = 5 cm.
To find: The radius of the Base of a cone.

Solution:

We Know that the volume of a cone 

Therefore

So the radius of the base of a cone is 12 cm.

Like this, you can apply the technique to make use of simple calculations.

I hope, you will definitely implement this technique to save your valuable time of Examination or test. The same logical thinking can be applied to improve your thinking level.

I simply don't want to tell you to go with this simple method of calculation. I want that slowly, you need to mold yourself and improve the thinking level. Only using this technique for solving some problem and getting good scores will not serve my purpose. I want that the entire world needs to improve the thinking skills so that everyone can invent so many beautiful things in the human life.

This is possible only with the help of students like you along with your parents.

Please come ahead and start this beautiful work for yourself and show to the world that we can also build the world with powerful thinking level as everyone is competent.

Anil Satpute

27-Simple Method to Solve Mensuration Problem (2)

One Student of 10^th Standard / 10^th grade asked me one question. I would like to share the method of solving the problem & its presentation with you. Please go through it & make the habit of applying the same technique to solve other problems & please share this technique with your friends & relatives. (Please note one thing that “By distributing our knowledge, it will boost & by keeping it only with us, it will reduce”).

   

The Problem is very simple. Please see the following Problem.

The Shape of the wooden Block is of the form of a frustum of a cone. Its Volume is 56980 cubic centimeters. If the radii of the top and bottom are 28 cm and 21 cm respectively, then find its height. (π = 22/7) (Note: Don’t use the calculator) Please see the following diagram.

Actually, this problem is very simple. But I would like to tell you the simple method of calculations & the proper method of presentation of this problem on paper.

First I would like to discuss the presentation, which will be useful for all the problems.

I would like to give numbers to each step. These numbers are to be written to the left of each step. Secondly, the equation numbers are to be given only the equations & to be written to the right of the equations. You can see the above problem which is solved by using the same technique.

Now we will discuss the basics of the problem.

Here we know that the volume of the frustum is 

Here we want to find height so 

So many students will solve this problem by taking the product of numerator & the product of Denominator & then take the division. I had done this problem with the different method of calculations which is very simple, saves your valuable time and authentic also.

First I will show how some student solves this problem & then I will explain its simple method of calculations so that you will definitely find the difference. I am sure that you will like this method in which you never require to do much more complicated calculations.

(Note: The description written in the box bracket in BLACK is only to understand the steps & not to be written in the solution of the problem)

So many Students may solve this problem by the following method

Note: You may observe fewer steps in this method but you will spend more time of Multiplication, Squaring the numbers 28 & 21 then the time of addition. This thing should be noted in my method given below.

Given: 1) V = 56980

             2) Radius of Top (Larger Circle) r₁ = 28 cm

             3) Radius of Bottom (Smaller Circle) r₂ = 21 cm

             4) π = 22/7

Find: Height h

                                        

1) We know that, 

  

   


[Here Students must take the complete square of 28 and 21 and the product of 28        and 21 as shown below

        


[Multiplying Numerator & denominator by 7 write down the addition of all three terms of the square bracket]

[Then the student will divide 56980 by 22                        and get 2590. and 784+441+588 = 1813]

     

                                      
[Then the student will divide numerator and denominator by 7 and write the results]

[Again dividing Numerator & Denominator by 7 and write the results]

[Divide Numerator & Denominator by 37 and write the results]
       
          h =  3 x 10

                               

So Height of the Wooden Block is h =   30 cm.

Please see the following simple method of calculations. It will definitely save your time & you will definitely like this method

Given: 1) V = 56980

           2) Radius of Top (Larger Circle) r₁ = 28 cm

           3) Radius of Bottom (Smaller Circle) r₂ = 21 cm

           4) π = 22/7

Find: Height h

1) We know that, 


         
[Here modify the term of the denomination using the formula [r₁² + r₂² – 2 x r₁ x r₂ = (r₁ - r₂) ^2. so we can write [r₁² + r₂² + (r₁ x r₂)] = r₁² + r₂² – 2 x r₁ x r₂ + 2x  r₁ x r₂ + r₁ x r₂] so we have]

 

[Here write [(r₁² + r₂² – 2 x r₁ x r₂) = (r₁ - r₂) ²  and 2 r₁ x r₂ + r₁ x r₂ = 3 r₁ x r₂]

            [Take 7 common from the denominator, we get]

                  [Take 7 common from the denominator, we get]

[Again divide Numerator & Denominator by 7, we get]

So Height of the Wooden Block is h =   30 cm.

Like this, you can also find so many ways of simple calculation methods and apply the same to other problems. At present, you may feel that these calculations are difficult. But while doing more practice of applying this technique, you may be more comfortable to use these simple calculation methods.

Anil Satpute

26-Parenting-13 Habit of Cleanliness will save our Environment

Click here for the next blog on parenting.

Morning to Evening, whenever we are outside our homes, when we watch here and there we observe that so many peoples throw garbage on the road. We know that the servicemen/servicewomen are allotted to clean the road. But we must think that why shouldn't we put our garbage in the proper places? There are so many dustbins placed on the roadside. If everyone will think this way, definitely the roads will be maintained properly and cleaner too. Secondly, the workers may clean the road systematically with less effort. I know that many places are clean. It is also true that so many peoples have this habit of putting the garbage in proper dustbins. I am not at all talking about these people. Here we need to explain to them the importance of cleanliness so that they will also follow the path of cleanliness. 

Right from a small age of our child, we need to teach them about cleanliness. Before and after lunch / Dinner / Breakfast, we need to clean our hands. Garbage and waste papers need to be put in the dustbins placed in every room. Keeping all our things in the proper places will reduce the work burden on others and it will improve the skills of your habit. Such important messages need to be given to the child to build the habits of good manners. 

I am not at all any big personality to teach you. I am a very ordinary person. These are my views which I would like to share with you and request you kindly help me to improve these skills in every one of us so that we all together can build the entire world with good attitudes. 

In a child, right from a small age, if we try to mold them in any shape definitely and quickly, they will get that shape and we need to get the benefit of the same to mold them with all the good things required for their better lifestyle. 

It looks very dirty if we see a group of highly qualified people going in a very rich car and throwing a used water bottle out on the road. Some people spit on the road or the staircases of the building. The persons throwing the rappers of chocolates outside the bus/train while traveling. These pictures will be observed in so many places. 

This is one part of our lifestyle that we need to build in the small mind of our child so that he/she will never do such type of blunders in the future. 

I am not at all pointing it out to any particular person or group. Once again I would like to clarify this point that we all together will build a beautiful world. I may need your valuable suggestions and help to motivate my own mind.  

Click here for the next blog on parenting.

ANIL SATPUTE

25-Magic Square-3 (using playing cards)

Click here for the next blog on the magic square.

Today we will see how to prepare the Magic Square from Playing Cards. This is a very interesting topic for my eBook "Miraculous World of Numbers". We know that there are 52 cards in the deck of Playing Cards. 12 cards are Face Cards (King - K, Queen - Q, Jack - J) each of 4 types (Spades, Diamonds, Hearts & Clubs) and 4 types of Ace cards. Like this, all together we have 16 Cards (12 Face Cards & 4 Ace Cards. Making use of these 16 Cards, we will prepare the Magic Square of 4 by 4 (16) houses.

Please see the following diagram in which we arranged these 16 cards in such a way that each row & column has each card of each type. In short, no card is repeated.

A) 1st Row contains 1) King of Heart, 2) Ace of Diamonds, 3) Queen of Club & 4) Jack of Spades.

B) 2nd Row contains 1) Queen of Spades, 2) Jack of Club  3) King of Diamonds & 4) Ace of Hearts.

C) 3rd Row contains 1) Jack of Diamonds 2) Queen of Heart  3) Ace of Spades & 4) King of Club.

D) 4th Row contains 1) Ace of Club  2) King of Spade  3) Jack of Heart & 4) Queen of Diamonds.

Now we will assign numbers to Ace (A), King (K), Queen (Q), Jack (J), Heart, Diamond, Club, and Spade as shown in the following diagram.

Now according to the places of the cards & the numbers allotted to these cards are placed at the respective house of the Magic Square.

The card in the 1st-row & 1st column is King of Heart so its associated Number will be 26 the Number 1st-row & 1st Column of the Magic Square will be 26 as shown in the following Diagram.

Similarly, the 1st-row & 2nd Column is Ace of Diamonds so its associated number is 15 so in Magic Square 15 is written at the 1st-row & 2nd Column, and so on

Here the addition of the numbers in Each Row, Each Column & both diagonals is 126.

In the above Diagram, We used 15, 16, 17, 18, 25, 26, 27, 28, 35, 36, 37, 38, 45, 46, 47, and 48 as the Numbers obtained from the Card Numbers. Now we give 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, and 16 as the Reduced Numbers to the above group and form the Magic square. It will be as shown below.

Here the addition of the numbers in Each Row, Each Column & both diagonals is 34.

Please put your comments about this blog & try some new creations.

I thank Mrs. Jyoti Anil Satpute for designing all the diagrams of my blogs & also of My forthcoming eBook "Miraculous World of Numbers.

Click here for the next blog on the magic square.

ANIL SATPUTE