How should the wire of 28 m be cut so that the combined area of the circle and square is as small as possible ?

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Application of Derivative 

A piece of wire 28 cm long is to be cut into two pieces. One piece is to be made into a circle and another into a square. How should the wire be cut so that the combined area of the two figures is as small as possible?

Let the wire be cut at a distance of  x meter  from one end. Therefore then two pieces of wire be x m and (28-x) m.

Calculate Dimension of Circle and Square


Now 1st part be turned into a square and  the 2nd part be be made into a circle.

Since 1st part of the wire is turned into square. then its perimeter will be x m. 
So using formula of perimeter of square , we can calculate side of the square = x/4 m

Calculate Areas of Circle and Square


 Therefore Area of square = (x/4)(x/4) sq m

                                     A1 = x2/16


And  when 2nd part of the wire is turned to circle, then its perimeter ( circumference ) will be 28 - x m. So using formula of perimeter of square , And if  "r" be  radius of the circle , Then
Circumference of circle =  2 Ï€ r =  (28-x)
 ∴  r = (28-x)/2Ï€

We know that Area of Circle A2   = Ï€ r2  

                                     A2 =  Ï€[(28-x)/2Ï€]2  

Express Areas in terms of Function

To find value/s of x


Now to find the value of x for which this function A(x) is maximum or minimum ,put A(x) = 0


To Test the Minimum Value of  Function


 Now we have the value of "x" on which either A(x) have maximum or minimum value . To check the maximum or minimum value we have to find A''(x) as follows






So A''(x) has positive value Therefore A(x) shall have maximum value at x = 112/(Ï€ + 4)
Hence two pieces of wire should be of length x m and (28-x) m

These pieces should be of length 112/(Ï€+4) and 28Ï€/(Ï€ + 4)


Verification



we can calculate the sum of these pieces , it must be 28 m

1st part     

   
112/(Ï€+4) = 112/{(22/7)+4}=112×7/50 = 784/50

2nd part 


28Ï€/(Ï€ + 4) = {28×22/7}/{(22/7)+4} = 88×7/50 = 616/50

Sum of Two Parts 


 112×7/50 + 28×7/50 = (784+616)/50
                                                                 
  = 1400/50= 28 m

My previous Post 

Don't forget to   read this posts

Quiz of  Mathematics For You 

 How to find area bounded by two circles 




Do not Forget to watch this video of same Problem

You can  clear your doubts if any after watching this video


Conclusion



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HOW TO SOLVE LINEAR EQUATIONS OF THREE VARIABLES BY MATRIX METHOD

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Discussed  short cut for inverse of matrix and solving linear equations of three variables .
Consider three linear equations of three variables .

 2x + 3y - 4z = 10

 3x - 2y + 4z = 12

  x  -  y  +  z  =  5
Changing these equations into Matrix Form like this

 AX = B
X =  A-1 B - - - - -(1)

where A is Matrix of 3×3 order , which consist of  Coefficients of  x ,y  and z respectively. X is the matrix of order 3×1 and whose elements are  variables  in given linear equations . B is the matrix of 3×1 and consist of  the constant terms on the right hand sides of all the equations such that


How to Find A-1 

To find A-1 ,we have to check the determinant value of Matrix A, if the Determinant value of Matrix A is non Zero then  A-1 Exists, otherwise not.

How to find Determinant  Det |A|

|A| = 2(2) + 3(1) - 4(-1)

 |A| = 2(2) + 3(1) - 4(-1)

 |A| = 4 + 3 +4

 |A| =  7
Since |A| is non Zero therefore A-1 Exists

To find Co factors of elements  and Ad joint Matrix of A

1st of  all  put all the elements of matrix A in 3 rows and 3 columns as written in matrix A then copy the 1st and 2nd columns as  4th and 5th columns , After this we have 3×5 arrangement as shown is 1st  figure given below, Now  complete the arrangement as 5×5 by copying 1st row and 2nd row as 4th and 5th row respectively. It can be seen in 2nd figure given below.

Find Co factors of A11  element and write it in C11   position


The element whose co factor is to be find out , is marked in red. and the co factor will be calculated by eliminated that row and column in which red coloured element is lying, Here co factor will be calculated by cross multiplication of  purple coloured four elements

Find Co factors of  A12  element and write it in C12   position

Here co factor will be calculated by cross multiplication of  purple coloured four elements

Find Co factors of  A13  element and write it in C13   position

Here co factor will be calculated by cross multiplication of  purple coloured four elements

Find Co factors of A21  element and write it in C21   position




Find Co factors of A22  element and write it in C22   position




Find Co factors of  A23  element and write it in C23   position




Find Co factors of A31  element and write it in C31   position


Find Co factors of A31  element and write it in C31   position



Find Co factors of A31  element and write it in C31   position


How to find values of  x , y and z


Therefore ad joint of Matrix A can be written as below


Now using the property of equality of two matrices ,
x   =   52/22   
y   =   -18/11 and 
z    =  -15/11

Verification 

we can check whether the values of x , y and z so calculated satisfies our system of linear equation by putting their values in one or all the given equations.

My previous Posts 

Don't forget to   read this posts

 Quiz of  Mathematics For You 

 How to find area bounded by two circles 




Do not Forget to watch this video of same Problem

You can  clear your doubts if any after watching this video





Conclusion

Thanks for visiting this website and spending your valuable time to read this post regarding HOW TO SOLVE LINEAR EQUATIONS OF THREE VARIABLES BY MATRIX METHOD , short cut for inverse of matrix .If you liked this post , don't forget to   share it with your friends to benefit them also ,we shall meet in next post , till then bye and take care......



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A QUIZ OF MATHEMATICS FOR YOU ,MATRICES QUESTION PAPERS

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A Quiz Of Mathematics For You 

There are Total 10 questions in all . Each question is assigned with 2 marks. Attempt all Question. Matrices question papers, determinant questions and answers, All the questions are from matrices and Determinants . Score will be Displayed at the end of Quiz.

TO CRACK  ANY COMPETITIVE EXAM VISIT THIS YOUTUBE CHANNEL FOR MATHS AND REASONING 

EXAM CRACKER

My previous Posts



FINAL WORDS


  Thanks for watching and responding to this quiz based on Matrices and Determinants on Mathematics. 


Appeal 


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HOW TO FIND AREA BOUNDED BY THREE LINES AND CIRCLES , AREA UNDER CURVES BY INTEGRATION METHOD

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How to find common area of three lines and one circles which are  intersecting at different  points with the help of an example.

 Given Lines and Curves

Consider one circle and three lines whose equations are  given below
( x - 1 )2 + y2 = 12            ......................(1)

       y = x                            .....................(2)

 y = -√3 (x-2)                      .....................(3)

 y = 0                                   .....................(4)



Let us draw these lines and circle in coordinate planes, We can compare the equation of circle  with standard form of circle to find  the coordinate of  centre of the circle is  (1,0)  and radius of both the circles is 1.


How To Draw Figure



HOW TO FIND AREA BOUNDED BY THREE LINES AND CIRCLES1st of all check whether these lines  intersect with circle or not . And if these lines intersect with each other or with circle then what are their coordinates of points of intersections.

Solve (1) and (2)


 Putting the value of  'y' from (2) in equation (1), we get
( x - 1 )2 + x2 = 12   

  ⇒  x2  + 12 -2×1×x +x2 = 12   

   ⇒  -2x +x2 = 0
    ⇒   2x(-1+x) = 0
Either  2x= 0 or (-1+x) = 0
          ⇒    x = 0 or x = 1
Now putting the values of x in (2) we get 
x = 0 when x = 0  and y = 1 when x = 1
Therefore points of intersection of (1) and (2) are
O( 0 , 0 ) and A( 1, 1 )


Solve (1) and (3)


Putting the value of  'y' from (3) in equation (1), we get
( x - 1 )2 + [-√3 (x-2)]2 = 12 
⇒  x 2 + 1 2  - 2x +  3 (x-2)2 = 1


 ⇒ x 2 + 1- 2x +  3 [ x 2 +4 - 4x] =1

  ⇒ x 2 + 1- 2x +  3x 2 + 12 - 12x -1 = 0 
  ⇒  4x 2 - 14x +12 = 0 

   ⇒  2x 2 - 7x + 6 = 0 ,    By Factorisation Method

    ⇒ 2x 2 - 4x - 3x  + 6 = 0 



  ⇒ 2x(x-2) -3( x - 2) = 0



    ⇒ (x-2)( 2x - 3) = 0

Either (x-2) = 0   or ( 2x - 3) = 0
   x = 2 and x = 3/2
To find the values of y , put both the values of  'x' in (3) .i.e. in  y = -√3 (x-2) 

when x = 2 ,    then  y = 0
and    x = 3/2 , then  y = √3 /2

Therefore points of intersection of (1) and (3) are
C( 2 , 0 ) and B( 3/2, √3 /2 )

Solve (2) and (3)


is the coordinate of Point of intersection of Line (2) and (3)


Same problem with the help of this Video ⇊




How to Find Required Area


Required Area = Shaded Area = Area of Î”OAL + Area of Curve ABMLA+ Area of Î” BCM
HOW TO FIND AREA BOUNDED BY THREE LINES AND CIRCLES
After simplification , we get  
HOW TO FIND AREA BOUNDED BY THREE LINES AND CIRCLES



My Previous Posts
 How to find area bounded by two circles 




Final words 



Thanks for visiting this website and spending your valuable time to read this post regarding how to find area bounded by three lines and circle  .If you liked this post , don't forget to   share it with your friends to benefit them also ,we shall meet in next post , till then bye and take care......


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HOW TO FIND COMMON AREA OF TWO PARABOLAS ,AREA UNDER TWO PARABOLAS , AREA UNDER CURVES

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How to find common area of two parabolas  , Area under two parabolas , Area of region bounded by two parabolas .


 Let us consider two parabolas whose equations are given by
y2 =  4ax  --------------  (1)
x2 =  4ay ----------------  (2)

To check whether these parabolas intersect with each others or not And if they intersect then what is/are their point/s of intersection.

How to find Points of Intersection

To find coordinate of points of intersection ,we have to solve equation (1) and (2)

Consider   eq (2) 
 x2 =  4ay 
⇒ y  = x2 /4a   ---------------(3)

 Putting the value of "y" in equation  (1) ,we get
HOW TO FIND COMMON AREA OF TWO PARABOLAS
Area under Curves
(x2 /4a)2 =  4ax
 ⇒    x4/16a2 =  4ax

⇒    x4=  64xa3
 ⇒   x4- 64xa3= 0

 Taking 'x' common 

 x(x3- 64a3) = 0
Either  x = 0    or x3- 64a3 = 0
⇒ x = 0    or    (x)3- (4a)3 = 0 
⇒ x = 0    or    (x)3- (4a)3 = 0 

 ⇒ x = 0    or    (x-4a)[ (x)2+ (4a)2 + (x)(4a)]   = 0 

 ⇒ x = 0    or    (x-4a) =  0  or  [ (x)2+ (4a)2 + (x)(4a)]   = 0 
⇒ x = 0    or    x =  4a  or   (x)2+ (4a)2 + (x)(4a)   = 0 

Since   x2+ 4a.x + 16a2   = 0   have no real  roots ,because its discriminant is negative, therefore this quadratic equation have complex roots. And these roots are rejected .

To find values of y

Now putting both  values of  "x"  in eq (3) i. e . y  = x2 /4a   ,we get
1st  put x = 0 
y = 0 / 4a = 0         ⇒ when x = 0 then y = 0
and put x = 4 
y =  (4a)2 /(4a)
y =  4a                    ⇒ when x = 4a then y = 4a

Hence two points of intersection of (1) and (2)   O(0,0) and A (4a , 4a) .
Now draw two parabolas using their points of intersections as drawn in given picture.

How to Find Required Area

Now to find the area enclosed between two Parabolas.
Required Area = shaded Area =Area OLAMO - Area ONAMO

HOW TO FIND COMMON AREA OF TWO PARABOLAS

Watch this video to remove your doubts if any




My Previous Posts
  How to find area bounded by two circles 




Final words 



Thanks for visiting this website and spending your valuable time to read this post regarding how to find area bounded by two parabolas  .If you liked this post , do share it with your friends to benefit them also ,we shall meet in next post , till then bye and take care......
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HOW TO FIND SAMPLE SPACE FOR TOSSING OF FIVE COINS

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Let us discuss the coin toss probability formula, sample space of tossing 1 coin , sample space of tossing 2 coins , sample space of tossing 5 coins , How to make a tree diagram.

To find the sample space for tossing of one  coin

HOW TO FIND SAMPLE SPACE FOR TOSSING OF one COIN
we know that there are two outcomes in tossing of coin . Out of two outcome one is H and second is T.
So  Put H and T in sample space 'S'
S = {H , T}

To find the sample space for tossing of two coins

HOW TO FIND SAMPLE SPACE FOR TOSSING OF two COINS
1st put  22   = 4  elements in a set of sample space,

1st character of 1st two elements must be H and 1st character of last two elements must be T .
2nd character of all the  elements must be H and T alternatively


 S={ HH , HT , TH , TT}

To find the sample space for tossing of three coins


1    Put  23  = 8  elements in a set of sample space,
HOW TO FIND SAMPLE SPACE FOR TOSSING OF FOUR COINS2    1st character of 1st four elements must be H and 1st character of last four elements must be T .

3   2nd character of 1st ,2nd ,5th and 6th  elements must be H and  2nd character of 3rd, 4th, 7th and 8th  elements must be T alternatively
4    Last character of all the  elements must be H and T alternatively.
Therefore sample  space  'S'  is given by




S = { HHH , HHT , HTH , HTT ,THH, THT , TTH , TTT }

To find the sample space for tossing of Four coins

1    Put  24  = 16  elements in a set ,
2    1st character of 1st eight elements must be H and 1st character of last eight elements must be T.

3   2nd character of 1st four and 9th to 12th  elements must be H and  2nd character of 5th, 6th , 7th and 8th and last four elements must be T alternatively.
4  3rd character for 1st , 2nd , 5th , 6th , 9th , 10th , 13th and 14th elements must be H and for remaining characters ( 3rd , 4th ,7th ,8th , 11th ,12th ,15th and 16th)  it must be T.
5    Last character of all the  elements must be H and T alternatively.

Hence sample space ' S' is given by 


S = { HHHH , HHHT , HHTH , HHTT , HTHH , HTHT , HTTH , HTTT , THHH , THHT , THTH , THTT , TTHH , TTHT , TTTH , TTTT }


To find the sample space for tossing of Five Coins

1    Put  25  = 32  elements in a set ,
2    1st character of 1st sixteen elements must be H and 1st character of last sixteen elements must be T.

3   2nd character of 1st eight and  17th to 24th  elements must be H and  2nd character of 9th to 16th and 25th to 32nd  elements must be T alternatively.

4  3rd character for 1st four , 9th to 12th , 17th to 20th , 25th to 28th elements must be H , And  3rd character for 5th to 8th , 13th to 16th , 21st to 24th and 29th to 32nd elements  must be T.

 5  4th character for 1st two ,5th and 6th ,9th and 10th , 13th and 14th , 17th and 18th, 21st and 22nd , 25th and 26th, 29th and 30th and 32nd must be H  And 3rd and 4th , 7th and 8th , 11th and 12th and 15th and 16th , 19th and 20th , 23rd and 24th , 27th and 28th , 31st and 32nd elements must be T.

6    Last character of all the  elements must be H and T alternatively.

Hence sample space ' S' is given by 
S = { HHHHH , HHHHT , HHHTH , HHHTT , HHTHH , HHTHT , HHTTH , HHTTT , HTHHH, HTHHT, HTHTH , HTHTT, HTTHH, HTTHT, HTTTH, HTTTT, THHHH , THHHT , THHTH , THHTT , THTHH , THTHT , THTTH , THTTT ,TTHHH , TTHHT, TTHTH , TTHTT, TTTHH, TTTHT , TTTTH , TTTTT }


Watch this video for finding sample space of coins

Final Words

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