A footpath of uniform width runs all around the inside of a rectangular field 54 m long and 35 m wide. If the area of the path is 420 m2, find the width of the path.

Given:


Length of field = 54 m


Breadth of field = 35 m


Let width of the path be x m



Area of field = Length × Breadth


= 54 m × 35 m


= 1890 m2


Therefore,


Length of field without path = 54 - (x + x)


= 54 - 2x


Breadth of field without path = 35 - (x + x)


= 35 - 2x


Therefore,


Area of field without path = Length without path × Breadth without path


= (54 - 2x) × (35 - 2x)


= 1890 – 70x – 108x + 4x2


= 1890 – 178x + 4x2


Now,


Area of path = Area of field - Area of field without path


420 = 1890 – (1890 – 178x + 4x2)


420 = 1890 – 1890 + 178x - 4x2


420 = 178x - 4x2


4x2 - 178x + 420 = 0


2x2 - 89x + 210 = 0


2x2 - 84x – 5x + 210 = 0


2x(x - 42) – 5(x – 42) = 0


(x - 42)(2x – 5) = 0


This gives us two equations,


i. x - 42 = 0


x = 42


ii. 2x – 5 = 0



Since, width of park cannot be more than breadth of field


Therefore, width of park = 42 m


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