In the centre of a rectangular lawn of dimensions 50m×40m, a rectangular pond has to be constructed so that the area of the grass surrounding the pond would be 1184m2. Find the length and breadth of the pond.
the fact that a rectangular pond has to be constructed in the centre of a rectangular lawn of dimensions 50m×40m So, the distance between pond and lawn would be same around the pond. Say x m.
Now, length of rectangular lawn (l1)=50m
and breadth of rectangular lawn (b1)=40m
∴ Length of rectangular pond (l2)=50−(x+x)=50−2x
and breadth of rectangular pond (b2)=40−(x+x)=40−2x
Also, area of the grass surrounding the pond =1184m2
∴ Area of rectangular lawn - Area of rectangular pond = Area of grass surrounding the pond.
l1×b1−l2×b2=1184(∴) area of rectangle = length×breadth
⇒ 50×40−(50−2x)(40−2x)=1184⇒2000−(2000−80x−100x+4x2)=1184⇒80x+100x−4x2=1184⇒4x2−180x+1184=0⇒x2−45x+296=0⇒x2−37x−8x+296=0[by splitting the middle term]
⇒x(x−37)−8(x−37)=0⇒(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 34 m and 24 m, respectively.