Mathematics

A footpath of uniform width runs around the inside of a rectangular field of 32 m long and 24 m wide. If the path occupies 208 m², find the width of the footpath.

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Let the width of footpath be x meters.

Length of field without path = (32 - 2x) m

Breadth of field without path = (24 - 2x) m

Area = Length × Breadth

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

⇒ 208 = (32)(24) - (32 - 2x)(24 - 2x)

⇒ 208 = 768 - (768 - 64x - 48x + 4x²)

⇒ 208 = -4x² + 112x

⇒ 4x² - 112x + 208 = 0

⇒ 4(x² - 28x + 52) = 0

⇒ x² - 26x - 2x + 52 = 0

⇒ x(x - 26) - 2(x - 26) = 0

⇒ (x - 2)(x - 26) = 0

⇒ x = 2 or 26

x ≠ 26 as it exceeds breadth.

∴ x = 2

Hence, width of the footpath = 2 m.

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