(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 $420m^2$, find the width of the path. 

Given:
Length of field = 54 m
Breadth of field = 35 m
Let x m be the width of the path.
Area of field $=Length\times Breadth$
$=54\times 35$
$=1890m^2$
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 $\times$ Breadth without path
                                           $=(54-2x)(35-2x)$
                                           $=1890-70x-108x+4x^2$
                                           $=4x^2-178x+\mathrm{1890...}(1)$

Area of path $=$ Area of field $-$ Area of field without path
$420=1890-(1890-178x+4x^2)$
(Using equation (1))
$420=178x-4x^2$
or $4x^2-178x+420=0$
or $2x^2-89x+210=0$
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.