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Question
A footpath of uniform width runs all around the inside of a rectangular field 54m long and 35 m wide. If the area of the path is `420m^2` , find the width of the path.
Solution
Area of the rectangular field =`54xx35=1890m^2`
Let the width of the path be x m. The path is shown in the following diagram:
Length of the park excluding the path=`(54-2x)m`
Breadth of the park excluding the path=`(35-2x)m`
Thus, we have:
Area of the path=`420m^2`
⇒`420=54xx35-(54-2x)(35-2x)`
⇒`420=1890-(1890-70x-108x+4x)^2`
⇒`420=4x^2+178x`
⇒`4x^2-178x+420=0`
⇒`2x^2-89x+210=0`
⇒`2x^2-84x-5x+210=0`
⇒`2x(x-42)-5(x-42)=0`
⇒`(x-42)(2x-5)=0`
⇒`x-42=0 or 2x-5=0`
⇒`x=42 or x=2.5`
The width of the path cannot be more than the breath of the rectangular field.
∴ `x=2.5 m `
Thus, the path is 2.5m wide.
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