(i) 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 420m2, find the width of the path.
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Solution
Given: Length of field = 54 m Breadth of field = 35 m Let x m be the width of the path. Area of field =Length×Breadth =54×35 =1890m2 Length of field without path =54−(x+x)=54−2x Breadth of field without path =35−(x+x)=35−2x Now, Area of field without path = Length without path × Breadth without path =(54−2x)(35−2x) =1890−70x−108x+4x2 =4x2−178x+1890...(1) Area of path = Area of field − Area of field without path 420=1890−(1890−178x+4x2) (Using equation (1)) 420=178x−4x2 or 4x2−178x+420=0 or 2x2−89x+210=0 On solving above equation, we have x=42 or x=5/2=2.5 As width of park cannot be more than breadth of field, so width of park is 2.5 m.