Question

hi there


maths legends please answer

Answer 1

It is given that the co ordinates of orthocenter H are ( 4 , b ) , centroid G are ( b , 2b - 8 ) and circumcenter S are ( - 4 , 8 ) .



We know that the ratio of HG and GS remains 2 : 1, where H is the orthocenter, S is the circumcenter and G is the centroid of the triangle.


Thus, in this case :
HG : GS = 2 : 1



Now,

= > HG = √{ ( 4 - b )² + ( b - 2b + 8 )² }

= > HG² = ( 4 - b )² + ( 8 - b )²

= > HG² = 16 + b² - 8b + 64 + b² - 16b

= > HG² = 2b² - 24b + 80 ---: ( 1 )



And,
= > GS = √{ ( b + 4 )² + ( 2b - 8 - 8 )² }

= > GS² = ( b + 4 )² + ( 2b - 16 )²

= > GS² = b² + 16 + 8b + 4b² + 256 - 64b

= > GS² = 5b² - 56b + 272 ---: ( 2 )



From above,

= > HG : GS = 2 : 1


Square on both sides,

= > HG² : GS² = 4 : 1

= > HG² = 4 GS²


Now, substituting the values of HS² and GS² from ( 1 ) and ( 2 ),

= > ( 2b² - 24b + 80 ) = 4( 5b² - 56b + 272 )

= > 2( b² - 12b + 40 ) = 4( 5b² - 56b + 272 )

= > b² - 12b + 40 = 10b² - 112b + 544

= > 0 = 10b² - b² - 112b + 12b + 544 - 40

= > 9b² - 100b + 500 = 0

$= > b = \dfrac{ - ( - 100) \pm \sqrt{( - 100) {}^{2} - 4(9 \times 504)} }{2 \times 9} \\ \\ \\ = > b = \dfrac{100 \pm \sqrt{10000 - 18144} }{18} \\ \\ \\ = > b = \dfrac{100 \pm \sqrt{ - 8144} }{18}$



The square root of any number does not exist in the real set, so b ≠ R( rational numbers ).



Therefore,
There is no value of b.

Hence,
Number of value of b is 0.
$\:$

Answer 2

Step-by-step explanation:

We  know  that,

Mass = Density × Volume

m = ρV

Multiplying  Both sides with g,

mg =  ρVg

Now, Here  mg  is the  weight(Force  Applied)  of  the System, Now, We  can Write it  as

W =  ρVg

Where  W  is  the  weight  of the  System

Case I:- When the  System is in Air,

W = 6000N

ρ₁(Iron) = 7.87 g/cm³ = 7.87 × 10³ kg/cm³

In this  case  we will calculate  only weight of  iron because  air  has  no weight.

W = ρ₁V₁g

6000 =  ρ₁V₁g

V₁ = 6000/ρ₁g

Case I:- When system is  in Water

Given that  weight of the sytem Cavities in this case is 4000N

We know  that  By  Principal of  Archimedes Weight = Volume  of the water displaced. (Supposing  that  iron system has  been completely submerged in water.)

If the  volume of  iron block+cavities = V,Then,

Weight reduced = ρ₂Vg

4000 = ρ₂Vg

V = 4000/ρ₂g

Hence,

Volume of  cavities = V - V₁

  =  4000/ρ₂g - 6000/ρ₁g

  = 2000/g(2/ρ₂ - 3/ρ₁)

  = 2000/g(2/(10³) - 3/7.87 × 10³)

  =   2/g(2- (3/7.78))

  =2/g( 2 - 0.38)

  = 2/g × 1.62

  = 3.24/9.8

  = 0.33 m³