There is a square field whose side is 44 m a square flower bed is prepared in its centre leaving a gravel path all around the flower bed the total cost of playing the flower pattern gravelling the part that two piece 2.75 and rs 1.50 per square metre respectively is rupees 4904 find the width of the gravel path
Answers
Answer: 4m
Step-by-step explanation:
Let the width of the gravel path = x m
Sides of the flower bed = (44 - x) m
Area of flower bed = (44-x)(44-x) = 1936 -88x+x^2
The cost of laying the flower bed = 2.75(1936 -88x+x^2) = 5324 - 242x + 2.75x^2
The area of the gravel path = 44x44 - (1936 -88x+x^2)
= 1936 - 1936 + 88x - x^2
= 88x - x^2
Cost of graveling = 1.5(88x - x^2) = 132x - 1.5x^2
Total cost = 5324 - 242x + 2.75x^2 + 132x - 1.5x^2
= 5324 - 110x + 1.25x^2
But the total cost = 4904
So 5324 - 110x + 1.25x^2 = 4904
re-arrange
1.25x^2 - 110x + 420 = 0
x^2 - 88x + 336 = 0
x^2 -84x - 4x + 330 = 0
x(x - 84) - 4(x - 84) = 0
(x - 4)(x - 84) = 0
Either x - 4 = 0 giving x = 4
Or x - 84 = 0 Giving x = 84
Since the side of the field is 44, we take x = 4
So, the width of the gravel path = 4m
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Answer: Width of the gravel path is 2 m
Step by step explanation:
To find: We have to find width of Gravel path.
Let the width of the gravel path be x
Each Sides of the square flower bed = (44 - 2x) m.
According to formula,
Area of the square field= side × side
Area of the square field= 44× 44 = 1936 m²
Area of flower bed = (44-2x)² = 1936 -176x+4x²
Area of the gravel path = area of the square field - area of the flower bed
= 1936 - (1936 - 176x + 4x²)
= 1936 - 1936 + 176x - 4x²
= 176 x - 4x²
Cost of laying the flower bed =( Area of the flower bed) × ( rate per square metre)
= (44-2x)² × 2.75
=( 275/100)(44-2x)²
= 11/4 × (44-2x) (44-2x)
= 11/4 ×2 (22 -x) 2 (22 -x)
= 11/4 ×4 (22 -x)²
= 11(22-x)²
We have to find cost of Graveling the path:
Cost of graveling the path =( Area of the path) × ( rate per square metre)
= (176x - 4x²) × 150/100
= 4 (44x -x²) × 3/2
= 4 × 3/2 × (44x -x²)
= 6× (44x -x²)
Total cost of laying the flower bed and graveling the path is ₹ 4904.
11(22-x)² + 6× (44x -x²) = 4904
= 11(484 - 44 x +x²) + (264x - 6x²)= 4904
= 11(484 - 44 x +x²) +6 (44x - x²)= 4904
= 5x² -220x + 5234 = 4908
= 5x² -220x +420 = 0
= x² - 44x +84= 0
= x² -42x -2x +84=0
= x( x-42) -2(x-42)=0
= (x-2) (x-42)= 0
x= 42
But x≠42, as the side of the square is 44 m.
Therefore x = 44-42 = 2
Hence, the width of the gravel path is 2 m.