There is a square field whose side is 44m . A square flower bed is prepared in its centre leaving a gravel path all around the flower bed. The cost of laying the flower bed and graveling the path at Rs 2.75 and Rs 1.50 per square meter, respectively, is Rs 4904. Find the width of the gravel path
Answers
Answer: Width of the gravel path is 2 m
Step by step explanation:
▪️ 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 Gravelling 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)
Given,
=> Total cost of laying the flower bed and gravelling 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= 2
Hence, the width of the gravel path is 2 m.
Let the width of the gravel path be x m.
Each Sides of the square flower bed = (44 - 2x) m.
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
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)
Given = the total cost of laying the flower bed and gravelling 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= 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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