32. A square flower bed is prepared in the centre of a square field, of side 40 m, leaving a gravel path all around the bed. The total cost of laying the flower bed and gravels at the path at the rate of Rs 2.75 and Rs 1.50 per sq. m respectively is Rs 4020. Find the width of the path.
Answers
Answer:
Let the width of the gravel path be x m.
Each Sides of the square flower bed = (40 - 2x) m.
Area of the square field= side × side
Area of the square field= 40× 40 = 1600 m²
Area of flower bed = (40-2x)² = 1600 -160x+4x²
Area of the gravel path = area of the square field - area of the flower bed
= 1600 - (1600 - 160x + 4x²)
= 1600 - 1600 + 160x - 4x²
= 160 x - 4x²
Cost of laying the flower bed =( Area of the flower bed) × ( rate per square metre)
= (40-2x)² × 2.75
=( 275/100)(40-2x)²
= 11/4 × (40-2x) (40-2x)
= 11/4 ×2 (22 -x) 2 (22 -x)
= 11/4 ×4 (22 -x)²
= 11(22-x)²
Cost of graveling the path =( Area of the path) × ( rate per square metre)
= (160x - 4x²) × 150/100
= 4 (44x -x²) × 3/2
= 4 × 3/2 × (44x -x²)
= 6× (44x -x²)
Given = the total cost of laying the flower bed and gravelling the path is ₹ 4904.
11(22-x)² + 6× (44x -x²) = 4020
= 11(484 - 44 x +x²) + (264x - 6x²)= 4020
= 11(484 - 44 x +x²) +6 (44x - x²)= 4020
= 5x² -220x + 5234 = 4020
= 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= 2 or x= 42
But x≠42, as the side of the square is 44 m. Therefore x= 2
Hence, the width of the gravel path is 2 m.
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