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

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Solution

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 $(l 2) = 50 - (x + x) = 50 - 2 x$
and breadth of rectangular pond $(b 2) = 40 - (x + x) = 40 - 2 x$
Also, area of the grass surrounding the pond = $1184 m^{2}$

∴ Area of rectangular lawn - Area of rectangular pond = Area of grass surrounding the pond.
$l 1 \times b 1 - l 2 \times b 2 = 1184 (∴)$ area of rectangle = $l e n g t h \times b r e a d t h$
⇒ $50 \times 40 - (50 - 2 x) (40 - 2 x) = 1184 \Rightarrow 2000 - (2000 - 80 x - 100 x + 4 x^{2}) = 1184 \Rightarrow 80 x + 100 x - 4 x^{2} = 1184 \Rightarrow 4 x^{2} - 180 x + 1184 = 0 \Rightarrow x^{2} - 45 x + 296 = 0 \Rightarrow x^{2} - 37 x - 8 x + 296 = 0$ [by splitting the middle term]
⇒ $x (x - 37) - 8 (x - 37) = 0 \Rightarrow (x - 37) (x - 8) = 0$
∴ $x = 8$
[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 = $50 - 2 x = 50 - 2 (8) = 50 - 16 = 34 m$
and breadth of pond = $40 - 2 x = 40 - 2 (8) = 40 - 16 = 24 m$
Hence, required length and breadth of pond are 34 m and 24 m, respectively.

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