there a square field whose four side is 44m. A square fliwer bed is prepared in its centre leaving a gravel path all round the flower bed. The total cost of laying the flower bed and the gravel path at Rs 2.75 and 1.50 per square meter, respectively, is Rs4904. Find width of the gravel path
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The answer is 2m
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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.