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For a hexagon, the area = 6 (1/2) (side)^2 *sqrt (3) /2

3/2  = 6 (1/2)(side)^2 sqrt (3) / 2

3/2 = 3side^2 sqrt (3) /2

1 = side^2 *sqrt (3)

1/sqrt 3 = side ^2

side =  sqrt  [ 1 /sqrt 3 ] ≈ .76

Surface area of a sphere =   4 pi r ^2      r = 10 units   is given 

                               S.A.   =   4 pi ( 10)^2 =  400 pi  units ^2 

LATERAL Surface area of sphere =   pi * 2r * h        but we know that   h = 2r 

                                                       = pi * 2r * 2r = 4 pi r^2

Surface area of a sphere = 4 pi r^2   

   so the difference between the sphere srface area and the lateral cylinder =   ZERO 

x^2 + 19x + 5  = 0

a^2 + a^3  

b^2 + b^3  

Sum of these roots =     (a^2 + b^2)  + (a^3 + b^3)  =   (a^2 + b^2)  + (a + b)  (a^2 + b^2 - ab)

Also

a^2 + a^3  = a^2(a + 1)        (1)

b^2 + b^3 =  b^2 ( b + 1)      (2)

Product of (1) , (2) = product of these  roots =  a^2*b^2* (a + 1) (b + 1)   =   (ab)^2 ( ab + (a + b)  + 1)

Product of roots in    x^2 + 19x + 5 = 0

ab  = 5

2ab = 10

Sum of roots =   a + b =   -19

Square both sides

a^2 + 2ab + b^2  =  361

a^2 + b^2 = 361  - 2ab

a^2 +  b^2  = 361 - 10

a^2 + b^2 = 351

Sum of roots  in our quadratic = 

(a^2 + b^2)  + (a + b) ( a^2 + b^2 - ab)  =

(351) + (-19) ( 351 - 5)  = -6223

Product of  roots  in our quadratic = 

(ab)^2 ( ab + (a + b) + 1)  =

(5)^2 ( 5  -19 + 1) =

(25) ( -13)  = 

-325

The quadratic is

x^2 + 6223x - 325

Angle OCD = 45

Angle COD  = 90

So....Angle ODC = 45

So triangle DCO  is 45  - 45 - 90

So  OD  = 10/sqrt 2  = OC  =  5sqrt 2

Area of triangle DCO   = (1/2) (5sqrt 2)(5sqrt 2)   = 25

Height of triangle DCO can be found as

25  = (1/2)CD * height

50 = 10 * height

50/10 =  height   = 5

Triangle ABO  similar to Triangle DCO

And the scale factor from triangle ABO  to Triangle DCO   = 4/10 =  2/5

So the height of triangle ABO  = (2/5) (height of tragle DCO)  = (2/5) (5)  = 2

So the trapezoid has a height of  5 + 2 =  7

And the bases are 10 and 4

Area =   (1/2) (7) (10 + 4)  =  49

What are the dimensions of the rectangle? Is the rectangle 9 x 11? 

(x^2 + 1/x)  * ( x^3  + 3x^2* x^4 + 3x (x^4)^2  + (x^4)^3 )  =

(x^2 + 1/x)  * ( x^3 + 3x^6 + 3x^9 + x^12)  =

( x ^3 + 1)  / x  * ( x^3 + 3x^6 + 3x^9 + x^12)  =

(x ^3 + 1) ( x^2 + 3x^5 + 3x^8 + x^11)  = 

(x ^2 + 3x^5 + 3x^8 + x^11) +  (x^5 + 3x^8 + 3x^11 + x^14)  =

x^14 + 4x^11  + 6x^8 + 4x^5 + x^2

x^2  + 12x + y^2 + 2y =  10    complete the square on x,y

x^2 + 12x  + 36 + y^2 + 2y + 1 =  10 +36 + 1

(x + 6)^2 + ( y + 1)^2  =   47

47  = r^2

Area  = 

pi* r^2

47 pi

Here :

https://web2.0calc.com/questions/coordinates_6772

3x^2 -5x -6 = 0

product of roots

ab = -6/3  =  -2

(ab)^2   = 4

2ab = -4

sum of  roots

a + b  =  5/3          square both sides

a^2 + 2ab + b^2  =  25/9 

a^2 - 4 + b^2  = 25/9

a^2 + b^2  = 25/9 + 4

a^2 + b^2  =  61/9        square both sides again

a^4 + 2 (ab)^2  + b^4  =  3721 / 81

a^4 + 2 (4) + b^4  = 3721/81

a^4 + b^4  + 8 = 3721/81

a^4 + b^4 = 3721/81 - 8  

a^4 + b^4  =  3073 /  81