Monday, March 4, 2013

27-Simple Method to Solve Mensuration Problem (2)

Blog-27                                                                             

Dear Students,

I delivered the lecture on Mathematical Fun.  It was delivered in Indian Language “Marathi”. If you understand Marathi, Please click here to see my Lecture on youTube.


Secondly to think more about the studies of Mathematics, you are requested to participate in Program of Studies for Students and Parents given in my previous Blog. (Blog-11)

One Student of 10th Standard / 10th 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 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 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 about 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
V = 1/3 π [r12 + r22 + (r1 x r2)] h

Here we want to find height so h = 3V/π [r12 + r22 + (r1 x r2)]. 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 different method of calculations which is very simple, saves your valuable time and authentic also.

First I will show how some student solve 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 following method

Note: You may observe less 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 bellow.

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
                                        π [r12 + r22 + (r1 x r2)] h
1) We know that, V = ----------------------------------
                                                          3
  
                                                    3V
     So                h = -------------------------------------       ------------- (1)
                                    π [r12 + r22 + (r1 r2)]
                                  
    
                3 x 56980
2)  So                    h = --------------------------------------------      
                                      22/7 x [282 + 212 + (28 x 21)]

[Here Students will do complete square of 28 & 21 and take the product of 28 & 21 as shown bellow]

                     3 x 56980
3)                           h = --------------------------------------------        
                                         22/7 x [784 + 441 + 588]

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

          21 x 56980
4)                           h = ---------------------
                                         22 x 1813  

[Then student will divide 56980 by 22 & get 2590 & write it as shown bellow]

       21 x 2590
5)                         h = ------------------
                                         1813

[Then student will divide Numerator & Denominator by 7 & write the results]

       21 x 370
6)                         h = ------------------
                                         259

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

        3 x 370
7)                         h = -----------------
                                        37
[Divide Numerator & Denominator by 37 & write the results]

      
8)                         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

                                    π [r12 + r22 + (r1 x r2)] h
1) We know that, V = ----------------------------------
                                                     3
  
                                                    3V
     So                h = -------------------------------------       ------------- (1)
                                      π [r12 + r22 + (r1 x r2)]

[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 + 2 x r1 x r2 + r1 x r2] so we have]

  
                                                             3V
                      h = ----------------------------------------------------------------------           
                               π [(r12 + r22 – 2 x r1 x r2) + 2 x r1 x r2 + r1 x r2]


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

                                                   3V
                           h = ----------------------------------------           
                                   π [(r1 - r2) 2 + 3 x r1 x r2]

              3 x 56980
2)                         h = --------------------------------------------
                                      22/7 [(28-21)2 + 3 x 28 x 21]

[Multiplying Numerator & denominator by 7, we get]

           3 x 7 x 56980
3)                         h = --------------------------------------------
                                      22 [(28-21)2 + 3 x 28 x 21]

[Writing (28 – 21) = 7 & 28 = (4 x 7) & 21 = (3 x 7), we get]


           3 x 7 x 56980
4)                         h = --------------------------------------------
                                      22 [(7)2 + 3 x 4 x 7 x 3 x 7]

[Re-arranging the term in the denominator]


           3 x 7 x 56980
                           h = --------------------------------------------
                                      22 [(7)2 + 3 x 4 x 3 x (7)2]


[Taking (7)2 Common from Box Bracket), we get]

          3 x 7 x 56980
5)                         h = --------------------------------------------
                                        22 x (7)2 [1+ (3 x 4 x 3)]



       3 x 7 x 56980
                           h = ----------------------------------
                                        22 x (7)2 [1+ (36)]

   3 x 7 x 56980
                            h = --------------------------
                                        22 x (7)2 [37]

[Divide Numerator & Denominator by 7 & we get]

     3 x 56980
6)                         h = -------------------------
                                        22 x 7 x 37

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

   3 x 8140
7)                         h = ---------------------
                                        22 x 37

[Divide Numerator & Denominator by 11 & we get]

   3 x 740
8)                         h = -----------------
                                        2 x 37

[Divide Numerator & Denominator by 2 & we get]


   3 x 370
9)                         h = -----------------
                                            37
[Divide Numerator & Denominator by 37 & write the results]

      
10)                         h 3 x 10
                              
So Height of the Wooden Block is h =   30 cm.


So dear Students,

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 this simple calculation methods.

Please participate in Program of Studies for Students and Parents given in my previous Blog. (Blog-11)