Find two positive numbers whose sum is 16 and sum of whose cube is Minimum

Show that of all the rectangles inscribed in a circle of given radius . The Square has maximum Area.

Solutions

Let ABCD be rectangle which is  inscribed in a given circle of radius ‘r’

And Let θ be the angle between side of rectangle and Diameter of given circle.

Therefore from right angled  Δ ABC ,

We have 
  AB  = AC cosθ          ∵ AC = 2r
Let A(x) be the area of Rectangle ABCD
∴ A(x) = AB × BC
    A = (2r cos θ)(2r sin θ )
    A =  4r² sin θ cos θ
    A = 2r²  (2sin θ cos θ)
    A = 2r²  (sin 2θ )

⇒ 2r² 2 (cos 2θ ) = 0 ,As r² is constant
⇒cos 2θ = 0
⇒cos 2θ =cos (π/2)
⇒ θ = π/4

 =4r²  (-2sin 2θ ) 

∴ A has Maximum value at θ = π/4

Find two positive numbers whose sum is 16 and sum of whose cube is Minimum

Solution

Let us consider two numbers x and 16- x .
Then transforming our problem to mathematical form which says “sum of whose cube”  as follows
A (x) =   x³ + (16 - x)³…….. (1)
Differentiating both sides w .r. t  “x” , we get

     X = 8
So  x  =  8 will be the 1^st required numbers if Double derivatives of A  w. r. t  ‘x’ comes to be positive at x = 8.
Differentiate (2)  w. r. t. ‘x’  .

My previous Posts

HOW TO SOLVE LINEAR EQUATIONS OF THREE  VARIABLES  BY  MATRIX  METHOD

 How to find area bounded by two circles 

 How to integrate integral with square root in numerator     

How to find area of the circles which is interior to the parabola

How to find common area of two parabolas.

FINAL WORDS

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

Appeal 

If you are a mathematician Don't forget to visit my Mathematics You tube channel ,Mathematics Website and Mathematics Facebook Page , whose links are given below

Maths Website , Facebook Page and You Tube Channel

Read More

HOW TO SOLVE LINEAR EQUATIONS OF THREE VARIABLES BY MATRIX METHOD

How to find solutions of system of linear equations of three variables with  the help of Matrix Method. In this post we shall  solve such types of system of linear equations in which variables lies in the denominators.  Let us solve three linear equations of three variables given below by Matrix Method.

These are not simple linear equations to solve. But these equations can be made simple by following substitution .

After substitution , these equations can be  reduced to following simple linear equations

2x + 3y + 4z = 3

3x - 4y + 5z = 5

  x + 2y - 3z = 6

To solve these equations by Matrix Method , Let us transform this set of linear equations to Matrix form

AX = B, and  X =A^-1 B -----> (4)

where A is matrix comprises the coefficients  x , y and z respectively as shown in the matrix given below , And B is the matrix consist of constant terms on right hand side of these equations and X is the matrix of variables in the linear equations.

1st of all we have to find A^{-1  and then product of the matrices} A^-1 and B  as mentioned in equation (4). But to find inverse of matrix A, its determinant value must be non zero   ( Note it) .

Let us find determinant value of Matrix A as follows :-

|A| = 2[(-4)(-3)–(5)(2)]-3[(3)(-3)-(5)(1)]+4[(3)(2)-(-4)(1)] 

|A| = 2[12-10]-3[-9-5]+4[6+4] 

|A| = 2[2] -3[-14]+4[10] 

|A| = 4 +42+40 

|A| = 86 

Since the value of Determinant of Matrix a is non zero.

  ⇒    A^-1  exists .

To find inverse ,we have to find ad joint of Matrix A . And to find Ad joint of any Matrix , we have to find Co factor Matrix . 

What is Co Factor Matrix

Co factor Matrix can be obtained by the co factors of all the elements of Matrix written in same place  where the element was written originally. 
Formula for Co factor of any element 

        C_ij = (-1)^i+jM_ij 

Co factor of A₁₁ 

Co factor of A₁₁   element can be calculated by eliminating 1st row and 1st column and solving the remaining determinant.

= (-1)¹⁺¹M₁₁ 

= 12 - 10 = 2

Co factor of A₁₂ 

Co factor of A₁₂   element can be calculated   by eliminating 1st row and 2nd column and solving the remaining determinant.

= (-1)¹⁺²M₁₂ 

= -1(-9 - 5 )= 14

Co factor of A₁₃ 

Co factor of A₁₃   element  can be  calculated   by eliminating 1st row and 3rd column and solving the remaining determinant.

= (-1)¹⁺³M₁₃ 

= 6 + 4 = 10

Co factor of A₂₁ 

Co factor of A₂₁   element can be calculated   by eliminating 2nd row and 1st column and solving the remaining determinant.

= (-1)²⁺¹M₁₁ 

= -1(- 9 - 8) = 17

Co factor of A₂₂ 

Co factor of A₂₂   element  can be calculated   by eliminating 2nd row and 2nd column and solving the remaining determinant.
= (-1)²⁺²M₂₂ 

= - 6 - 4 = -10

Co factor of A₂₃ 

Co factor of A₂₃   element can be calculated   by eliminating 2nd row and 3rd column and solving the remaining determinant.

= (-1)²⁺³M₂₃ 

= -1(4 - 3) = -1

Co factor of A₃₁

Co factor of A₃₁   element can be calculated   by eliminating 3rd row and 1st column and solving the remaining determinant.

= (-1)³⁺¹M₃₁ 

 

= 15 + 16 = 31

Co factor of A₃₂

Co factor of A₃₂   element  can  be calculated   by eliminating 3rd row and 2nd column and solving the remaining determinant.

= (-1)³⁺²M₃₂ 

= -1(10 - 12) = 2

Co factor of A₃₃

Co factor of A₃₃   element  can  be calculated   by eliminating 3rd row and 3rd column and solving the remaining determinant.

= (-1)³⁺³M₃₃ 

= - 8 - 9 = -17

Co Factor Matrix of A

Now write all the co factors calculated in Matrix Form as follows

Ad joint Matrix of A

To find the Ad Joint of  Co factor Matrix, transform 1st row to 1st column, 2nd row to 2nd column  and 3rd rows to 3rd column as follows 

Formula to Find   A^-1 

To Find the value of  A^-1 ,Putting the value of Adj A in the formula given below.

Find values of x ,y and z

Multiplying both the matrices which are on the right side.

Simplifying the matrix so obtained

Dividing each element by 86 to get matrix of 3×1 (i. e. Perform scalar multiplication of matrix ).

Using the equality of two matrices .We can write the values of x, y  and z respectively like this

 x =277/86  , y = 4/86  and z = -77/86

Now the values of P, Q and R can be calculated by putting the values of x , y and z in equations (1) , (2 ) and (3) respectively.

P = 86/277  ,Q = 86/4   and R = -86/77

Don't Forget to Watch this Video of same Method

Verification of solution

Putting values of P , Q and R in given  equations , these values must satisfies  three given equations

From 1st equation

Similarly other two equations can be verified

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

My Previous Posts

Don't forget to   read this posts

Quiz of  Mathematics For You 

 How to find area bounded by two circles 

 How to integrate integral with square root in numerator     

How to find area of the circles which is interior to the parabola

How to find common area of two parabolas.

Conclusion

I have discussed the method of solving the System of linear Equations with the help of Matrix Method , method to find inverse of a matrix ,How to find inverse of 3×3 Matrix,how to solve determinant,how to solve system of equation by matrix, matrix method of solving system of equations of three variables,If you liked the post Don't  forget to share it with your friends , And in case of any improvement please make use of Comment Box . 

Appeal

If you are a mathematician Don't forget to visit my Mathematics You tube channel , Mathematics Website and Mathematics Facebook Page , whose links are given below

Maths Website , Facebook Page and You Tube Channel

Read More

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

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 π r =  (28-x)

 ∴  r = (28-x)/2π

We know that Area of Circle A₂   = π r²  

                                     A₂ =  π[(28-x)/2π]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/(π + 4)
Hence two pieces of wire should be of length x m and (28-x) m

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

^Verification

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

^{1st part}     

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

2nd part 

28π/(π + 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 

 How to integrate integral with square root in numerator     

How to find area of the circles which is interior to the parabola

How to find common area of two parabolas.

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 should the wire of 28 m be cut so that the combined area of the circle and square is as small as possible , s .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......

If you are a mathematician Don't forget to visit my Mathematics You tube channel ,Mathematics Website and Mathematics Facebook Page , whose links are given below

Maths Website , Facebook Page and You Tube Channel

Read More

HOW TO SOLVE LINEAR EQUATIONS OF THREE VARIABLES BY MATRIX METHOD

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 

 How to integrate integral with square root in numerator     

How to find area of the circles which is interior to the parabola

How to find common area of two parabolas.

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......

If you are a mathematician Don't forget to visit my Mathematics You tube channel ,Mathematics Website and Mathematics Facebook Page , whose links are given below

Maths Website , Facebook Page and You Tube Channel

Read More