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.

Answer 1

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.