Step-by-step explanation:

area of sq field-44²

width of gravel= x

width of sq bed =(44-2x)

area of square bed=(44-2x)²=1936-176x²+4x²

area of gravel=ar of sq field - ar of sq bed

=(44)²-(44-2x)²=1936 - [1936-176x+4x²] = -4x²+176

cost of gravel=area×1.5= -6x²+246

cost of field= area ×2.75=5324-484x+11x²

total cost= -6x²+246+5324-484x+11x²=4904

=5x²-220x+420

=x²-44x+84

=x²-42x-2x+84

=(x-42)=0 or x-2=0

if x=42then 2x<<44 not possible

therefore x=2 width of gravel path

sq field = 40

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.