Question

Nonsense/No explanation = report

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Answer 1

Answer:

solution

$area \: of \: pool \\ length \times breadth = (2x + 5)(2x - 1)m ^{2} \\ = (4x ^{2} + 8x - 5)m ^{2} \\ (4x ^{2} + 8x - 5)m ^{2} \times (x - 2)m \\ (4x ^{3} + 8x ^{2} - 5x - 8x ^{2} - 16x + 10 )m ^{3} \\ = (4x ^{3} - 21x + 10)m ^{3}$

Answer 2

Given :-

The dimensions of a pool are as follows

• Length = 2x + 5 metres
• Width = 2x - 1 metres
• Height = x - 2 metres

To Find :-

The value of ' x '

Solution :-

Before starting the answer , let's recall ;

Consider a cuboid with length , beadth/width and height l , b and h respectively . Then Volume of the cuboid is given by :-

• $\sf Volume = l \times b \times h$

And the discriminant of a quadratic equation " ax² + bx + c = 0 " is given by :-

• D = b² - 4ac

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Now , as in this question . Let us assume that ;

• Length of pool = l = 2x + 5 metres
• Width of pool = b = 2x - 1 metres
• Height of pool = h = x - 2 metres

Now , According to question :-

$\quad \leadsto \quad \sf l \times b \times h = 182$

${ : \implies \quad \sf ( 2x + 5 ) ( 2x - 1 ) ( x - 2 ) = 182 }$

${ : \implies \quad \sf \bigg\{ ( x - 2 ) ( 2x - 1 ) \bigg\} ( 2x + 5 ) = 182}$

${ : \implies \quad \sf \bigg\{ x ( 2x - 1 ) - 2 ( 2x - 1 ) \bigg\} ( 2x + 5 ) = 182}$

${ : \implies \quad \sf \bigg\{ 2x² - x - 4x + 2 \bigg\} ( 2x + 5 ) = 182 }$

${ : \implies \quad \sf (2x+5)(2x² - 5x + 2 ) = 182}$

${ : \implies \quad \sf 2x ( 2x² - 5x + 2 ) + 5 ( 2x² - 5x + 2 ) = 182}$

${ : \implies \quad \sf 4x³ - 10x² + 4x + 10x² - 25x + 10 = 182}$

${ : \implies \quad \sf 4x³ - \cancel{10x²} + \cancel{10x²} - 21x + 10 - 182 = 0 }$

${ : \implies \quad \sf 4x³ - 21x - 172 = 0 \quad \qquad \bf -----(i)}$

Now , let us assume that ;

• p ( x ) = 4x³ - 21x - 172

Now , use hit and trial method to find factor of

p ( x ) ;

At x = 1 ,

${ : \implies \quad \sf p ( 1 ) = 4 \times ( 1 )³ - 21 \times 1 - 172 }$

${ : \implies \quad \sf p ( 1 ) = 4 - 21 - 172 }$

${ : \implies \quad \sf p ( 1 ) = 4 - 151 }$

${ : \implies \quad \sf p ( 1 ) = - 147 \neq 0 }$

If we put , x = 4 ;

${ : \implies \quad \sf p ( 4 ) = 4 \times ( 4 )³ - 21 \times 4 - 172 }$

${ : \implies \quad \sf p ( 4 ) = 4 \times 64 - 84 - 172 }$

${ : \implies \quad \sf p ( 4 ) = 256 - 256 }$

${ : \implies \quad \sf p ( 4 ) = \cancel{256} - \cancel{256} }$

${ : \implies \quad \bf \therefore \quad p ( 4 ) = 0 }$

Here , p ( 4 ) = 0 implies that , ( x - 4 ) is a factor of p ( x ) ;

Now , Divide p ( x ) by ( x - 4 ) { See the attachment 2 }

From their we have ;

${ \quad \leadsto \quad \sf p ( x ) = ( x - 4 ) ( 4x² + 16x + 43 ) }$

Now , as p ( x ) = 0 { $\because$ By ( i ) }

${ : \implies \quad \sf ( x - 4 ) ( 4x² + 16x + 43 ) = 0 }$

Case I :- When ( x - 4 ) = 0

${ : \implies \quad \bf \therefore \quad x = 4 }$

Case II :- When ( 4x² + 16x + 43 ) = 0

Calculate Discriminant to find nature of roots :-

${ : \implies \quad \sf D = ( 16 )² - 4 \cdot 4 \cdot 43 }$

${ : \implies \quad \sf D = 256 - 688}$

${ : \implies \quad \bf \therefore \quad D = - 432}$

Here , D < 0 , which implies values of x will be imaginary . So, this is not possible for length

Henceforth , The Required Answer is 4 :)