Question

In the centre of a rectangular lawn of dimensions $50m\times 40m$, a rectangular pond has to be constructed so that the area of the grass surrounding the pond would be 1184 $m^2$ [see figure]. Find the length and breadth of the pond.

Answer 1

Given that a rectangular pond has to be constructed in the centre of a rectangular lawn of dimensions $50m\times 40m$ Let the width of the lawn be x metres.


Now, length of rectangular lawn $\left(l_1\right)=50m$
and breadth of rectangular lawn $\left(b_1\right)=40m$
$\therefore$ Length of rectangular pond $\left(l_2\right)=50-(x+x)=50-2x$
and breadth of rectangular pond $\left(b_2\right)=40-(x+x)=40-2x$
Also, area of the grass surrounding the pond = 1184 $m^2$
$\therefore$ Area of rectangular lawn - Area of rectangular pond = Area of grass surrounding the pond.
$l_1\times b_1-l_2\times b_2=1184$$\left[\right(\therefore )areaofrectangle=length\times breadth]$
$\Rightarrow 50\times 40-(50-2x)(40-2x)=1184\Rightarrow 2000-(2000-80x-100x+4x^2)=1184\Rightarrow 80x+100x-4x^2=1184\Rightarrow 4x^2-180x+1184=0\Rightarrow x^2-45x+296=0\Rightarrow x^2-37x-8x+296=0$ [by splitting the middle term]
$\Rightarrow x(x-37)-8(x-37)=0\Rightarrow (x-37)(x-8)=0$
$\therefore$ 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]
$\therefore$ 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.