A foot path of uniform width runs all around inside of a rectangular field 45m long and 36m wide. If the area of the path is 234m^2,find the width of the path
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Solution
ASsuming width of path to be 'x'm 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 = 1.5m