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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. - Mathematics

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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.

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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.

Concept: Area of Circle
  Is there an error in this question or solution?

APPEARS IN

RS Aggarwal Secondary School Class 10 Maths
Chapter 17 Perimeter and Areas of Plane Figures
Q 12

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