# The footpath of uniform width runs all around inside of a rectangular field 45m long and 36m wide if the area of the path is 234m find width

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Therefore, exclusive of the footpath:

=> length of field = 45-x-x = (45-2x)m

=> width of field = 36-x-x = (36-2x)m

Total area (exclusive of path)

= (45-2x)(36-2x)

= (1620-90x-72x+4x^2)

= (4x^2-162x+1620) sq.m

Total area (inclusive of path)

= (45)(36)

= 1620 sq.m

Area of footpath = 234m

Therefore,

Area of footpath = Area exclusive of path - Area inclusive of path

=> 234 = 1620-(4x^2-162x+1620)

=> 234 = -(4x^2-162x)

=> -234 = 4x^2-162x (Multiplying equation with -1)

=> 0 = 4x^2-162x+234

=> 0 = 2x^2-81x+117 (Dividing equation by 2)

=> 0 = 2x^2-78x-3x+117

=> 0 = 2x(x-39)-3(x-39)

=> 0 = (2x-3)(x-39)

=> 2x-3 = 0 or x-39 = 0

=> x = 1.5m or x = 39m

Discarding x = 39m, because that's longer than the breadth of the field.

Therefore, width of path =