In the centre of a rectangular lawn of dimensions 50 m×40 m, a rectangular pond has to be constructed so that the area of the grass surrounding the pond would be 1184 m2 [see figure]. Find the length and breadth of the pond.
34 m and 24 m
Given that a rectangular pond has to be constructed in the centre of a rectangular lawn of dimensions 50m×40m Let the width of the lawn be x metres.
Now, length of rectangular lawn (l1)=50 m
and breadth of rectangular lawn (b1)=40 m
∴ 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 = 1184 m2
∴ 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 = 34 m
and breadth of pond = 40 - 2x = 40 - 2(8) = 40 - 16 = 24 m
Hence, required length and breadth of pond are 34 m and 24 m, respectively.