In the center 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.

Open in App

Solution

let ABCD be rectangular lawn and EFGH be rectangular pond.
let x m be the width of grass area, which is same around the pond.

Given, length of lawn = 50m.
Width of lawn = 40m.
length of pond $= (50 - 2 x) m$
Breadth of pond $= (40 - 2 x) m$
Also given,
Area of grass surround the pond $= 1184 m^{2}$
$∴$ Area of rectangular lawn = Length × Breadth
$= 50 \times 40 = 2000 m^{2}$
$∴$ Area of pond = length × Breadth
$= (50 - 2 x) \times (40 - 2 x)$
$= (2000 - 80 x - 100 x + 4 x^{2})$
$= (2000 - 180 x + 4 x^{2}) m^{2}$
Area of grass surrounding the pond = (Area of rectangular lawn - Area of pond)
$\Rightarrow 1184 = 2000 - (2000 - 180 x + 4 x^{2})$
$\Rightarrow 2000 - 2000 + 180 x - 4 x^{2} = 1184$
$\Rightarrow 4 x^{2} - 180 x + 1184 = 0$
$\Rightarrow x^{2} - 45 x + 296 = 0$
$\Rightarrow x^{2} - 37 x - 8 x + 296 = 0$
$\Rightarrow x (x - 37) - 8 (x - 37) = 0$
$\Rightarrow (x - 8) (x - 37) = 0$
$\Rightarrow x - 37 = 0 o r x - 8 = 0$
$x = 37 o r x = 8$
Here, $x = 37$ is not possible. As length of pond becomes negative i.e., $50 - (2 \times 37) = - 24$
[which is not possible]
Thus, $x = 8$ is acceptable.
Therefore, length of pond $= 50 - (2 \times 8) = 50 - 16 = 34 m$
Breadth of pond $= 90 - (2 \times 8) = 90 - 16 = 74 m$
Thus the length and breadth of the pond are 34m and 24m respectively.

Suggest Corrections

Similar questions

In a rectangular park of dimensions 50m×40m, a rectangular pond is constructed so that the area of grass strip of uniform width surrounding the pond would be 1184m².Find the length and breadth of the pond.

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 $m^{2}$ [see figure]. Find the length and breadth of the pond.