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 420 m2, find the width of the path.
Given:
Length of field = 54 m
Breadth of field = 35 m
Let width of the path be x m
Area of field = Length × Breadth
= 54 m × 35 m
= 1890 m2
Therefore,
Length of field without path = 54 - (x + x)
= 54 - 2x
Breadth of field without path = 35 - (x + x)
= 35 - 2x
Therefore,
Area of field without path = Length without path × Breadth without path
= (54 - 2x) × (35 - 2x)
= 1890 – 70x – 108x + 4x2
= 1890 – 178x + 4x2
Now,
Area of path = Area of field - Area of field without path
⇒ 420 = 1890 – (1890 – 178x + 4x2)
⇒ 420 = 1890 – 1890 + 178x - 4x2
⇒ 420 = 178x - 4x2
⇒ 4x2 - 178x + 420 = 0
⇒ 2x2 - 89x + 210 = 0
⇒ 2x2 - 84x – 5x + 210 = 0
⇒ 2x(x - 42) – 5(x – 42) = 0
⇒ (x - 42)(2x – 5) = 0
This gives us two equations,
i. x - 42 = 0
⇒ x = 42
ii. 2x – 5 = 0
Since, width of park cannot be more than breadth of field
Therefore, width of park = 42 m