❣ 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
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!
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 .