A foot path of uniform width runs all around inside of a rectangular field 45 m long and 36 m wide. If the area of the path is 234 m², find the width of the path.

Consider ABCD as a rectangular field having, length = 45 m and breadth = 36 m.

Let x meters be the width of foot path.

We know that,

Area = length × breadth

From figure,

Area of path = Area of rectangle ABCD - Area of rectangle PQRS

Substituting the values we get,

Area of path = (AB × BC) - (PQ × QR)………(1)

From figure,

PQ = AB - x - x = (45 - 2x) m,

QR = BC - x - x = (36 - 2x) m.

Substituting the values in equation (1) we get,

⇒ 234 = (45 × 36) - (45 - 2x) (36 - 2x)

⇒ 234 = 1620 - [45(36 - 2x) - 2x(36 - 2x)]

⇒ 234 = 1620 - (1620 - 90x - 72x + 4x²)

⇒ 234 = 1620 - 1620 + 90x + 72x - 4x²

⇒ 234 = 162x - 4x²

⇒ 4x² - 162x + 234 = 0

⇒ 4x² - 156x - 6x + 234 = 0

⇒ 4x(x - 39) - 6(x - 39) = 0

⇒ (4x - 6)(x - 39) = 0

⇒ 4x - 6 = 0 or x - 39 = 0

⇒ 4x = 6 or x = 39

⇒ x = $\dfrac{6}{4} = \dfrac{3}{2}$ = 1.5 or x = 39

Since, width of path cannot be greater than breadth of field,

So, x ≠ 39 m.

Hence, width of path = 1.5 m.