In the centre of a rectangular lawn of dimensioms `50 mxx40 m` a rectangular pond has to be constructed so that the area of the grass surroundng the pond would be `1184 m^2` Find the length and breadth of the pond

Explanation

Given that a rectangular pond has to be constructed in the centre of a rectangular lawn of dimensions `50mxx40m,`. So the distance between pond and lawn would be same around the pond. Say `xm`

Now length of rectangular lawn `(l_(1))=50m`
and breadth of rectangular `(b_(1))=40m`
`:.` Length of rectangular pond `(l_(2))=50-(x+x)=50-2x`
and breadth of reactangular pond `(b_(2))=40-(x+x)=40-2x`
Also, area of the grass surrounding the pond `=1184m^(2)`
`:.` Area of rectangular lawn `-` Area of rectangular pond `=` Area of grass surrounding the pond
`l_(1)xxb_(1)-l_(2)xxb_(2)=1184` [ `:'` area of rectangle `=` length `xx` breadth]
`implies 50xx40-(50-2x)(40-2x)=1184`
`implies 2000-(2000-80x-100x+4x^(2))=1184`
`implies 80x+100x-4x^(2)=1184`
`implies4x^(2)-180x+1184=0`
`implies x^(2)-45x+296=0`
`implies x^(2)-37x-8x+296=0` [by splitting the middle term]
`implies (x(x-37)-8(x-37)=0`
`implies (x-37)(x-8)=0`
`:. x=8`
[At `x=37`, length and breadth of pond are `-24` and `-34`, respectively but length and breadth cannot be negative. So `x=37` cannot be possible]
`:.` Length of pond `=50-2x=50-2(8)=50-16=34m`
and breadth of pond `=40-2x=40-2(8)=40-16=24m`
Hence required length and breadth of pond are `34n` and `24m` respectvely.