Math, asked by FuturePoet, 1 year ago

❣ QUESTION on Quadratic Equation in word problems on Mensuration ❣

There is a square field whose sides is 44 m . A square flower bed is prepared in it's centre leaving a gravel path all round the flower bed . The total Cost of laying the flower bed and gravelling the path at $2.75 and $1.50 per square metre , respectively is $4904 . Find the width of the gravel path ?

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Answers

Answered by siddhartharao77
83

Answer:

2 m

Step-by-step explanation:

Given, Side of square field = 44 m.

Then, area of square field = 44² = 1936 m².

Let the width of the gravel = 'x' metres.

Then, Each side of the square flower bed = (44 - 2x) metres.

Area of flower bed = (44 - 2x)² m².

∴ Area of the gravel path = Area of the field - area of flower bed

⇒ 1936 - (44 - 2x)²

⇒ 1936 - 1936 - 4x² + 176x

⇒ (176x - 4x²) m².

Cost of lying the flower bed:

= (44 - 2x)² * 2.75 m²

⇒ (1936 + 4x² - 176x) * 2.75

⇒ 5324 + 11x² - 484x

Cost of graveling the path:

= (176x - 4x²) * 1.5

= 264x - 6x²

Given that total cost is 4904.

⇒ 5324 + 11x² - 484x + 264x - 6x² = 4904

⇒ 5x² - 220x + 420 = 0

⇒ x² - 44x + 84 = 0

⇒ x² - 42x - 42x + 84 = 0

⇒ x(x - 42) - 2(x - 42) = 0

⇒ (x - 2)(x - 42) = 0

⇒ x = 2, 42{Side of the square is 44 m}

⇒ x = 2.

Therefore, Width of the gravel path = 2 m.

Hope it helps!


Anonymous: Great answer :)
siddhartharao77: Great thanks dude!
siddhartharao77: Great thanks bro
siddhartharao77: Thanks
Answered by Anonymous
78

Question :

There is a square field whose sides are 44 m . A square flower bed is prepared in it's centre leaving a gravel path all round the flower bed . The total Cost of laying the flower bed and graveling the path at $2.75 and $1.50 per square metre , respectively is $4904 . Find the width of the gravel path ?

Answer:

2 m

Step-by-step explanation:

Let the width of the gravel be x .

Area of square field

Area = ( side )²

        = ( 44 m )²

        = 1936 m²

Side of the square flower field

Side will be 44 m - x - x

                 = 44 m - 2 x

Area of flower bed

Area = ( side )²

⇒ area = ( 44 - 2 x )²

Area of gravel path

Area = ( area of square - area of flower bed )

        = 1936 - ( 44 -  2 x )²

SHORTCUT

              = 44² - ( 44 - 2 x )²

              = ( 44 - 44 + 2 x )( 44 + 44 - 2 x )

              = 2 x ( 88 - 2 x )

              = 176 x - 4 x²

Laying the  gravel path

Cost = area ( in m² ) × cost per 1 m²

        = $ 1.50 × ( 176 x - 4 x² )

        = 264 x - 6 x²

Laying the flower bed

Cost = area ( in m² ) × cost per 1 m²

        = $ 2.75 × ( 44 - 2 x )²

   

TOTAL COST

Total cost = 2.75 ( 44 - 2 x )² + 264 x - 6 x² = 4904

⇒ 2.75 ( 1936 + 4 x² - 176 x ) - 6 x² = 4640

⇒ 5324 + 11 x² - 484 x - 6 x² - 4640 = 0

⇒ 5 x² - 220 x + 420 = 0

⇒ 5 x² - 10 x - 210 x + 420 = 0

⇒ 5 x ( x - 2 ) - 210 ( x - 2 ) = 0

⇒ ( 5 x - 210 )( x - 2 ) = 0

x = 2

5 x = 210

⇒ x = 210/5

⇒ x = 42

But we cannot take x as 42.

44 - 2 x = 44 - 2 ( 42 ) but side cannot be - ve.

The width of the gravel path is  2 m .

I am not scared of moderator's accounts .


Anonymous: Superb answer :)
Anonymous: thenku bhai ^^"
FuturePoet: Great answer Sis!
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