Question
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 $1184m^2$. Find the length and breadth of the pond.
✓
Answer
Pond and lawn both are rectangular. Pond is inside the lawn.
Let the length of pond = (50 - 2x)m
and the breadth of pond = (40 - 2x)m
But, Area of grass around the pond $= 1184m^2$
⇒ Area of Lawn - Area of Pond = 1184
$\Rightarrow 50 \times 40 - (50 - 2x)(40 - 2x) = 1184$
$\Rightarrow 2000 - (2000 - 100x - 80x + 4x^2) - 1184 = 0$
$\Rightarrow 2000 - (2000 - 180x + 4x^2) - 1184 = 0$
$\Rightarrow 2000 - 2000 + 180x - 4x^2 - 1184 = 0$
$\Rightarrow 4x^2 - 180x + 1184 = 0$
$\Rightarrow x^2 - 45x + 296 = 0$
$\Rightarrow x^2 - 37x - 8x + 296 = 0$
$\Rightarrow x(x - 37) - 8(x - 37) = 0$
$\Rightarrow (x - 37)(x - 8) = 0$
$\Rightarrow x - 37 = 0 or x - 8 = 0$
$\Rightarrow 3 = 37 or x = 8$
When x = 37, then
the length of pond = 50 - 2 × 37
$= 50 - 74$
$= -24m$
Length cannot be negative. So, x = 37 is rejected.
When x = 8, then
the length of pond = 50 - 2x
$= 50 - 2 × 8$
$= 50 - 16$
= 34m
and the breadth of the pond
$= 40 - 2x$
$= 40 - 2 × 8$
$= 40 - 16$
$= 24m$
Hence, the length and breadth of the pond are 34m abd 24m respectively.
Let the length of pond = (50 - 2x)m
and the breadth of pond = (40 - 2x)m
But, Area of grass around the pond $= 1184m^2$
⇒ Area of Lawn - Area of Pond = 1184
$\Rightarrow 50 \times 40 - (50 - 2x)(40 - 2x) = 1184$
$\Rightarrow 2000 - (2000 - 100x - 80x + 4x^2) - 1184 = 0$
$\Rightarrow 2000 - (2000 - 180x + 4x^2) - 1184 = 0$
$\Rightarrow 2000 - 2000 + 180x - 4x^2 - 1184 = 0$
$\Rightarrow 4x^2 - 180x + 1184 = 0$
$\Rightarrow x^2 - 45x + 296 = 0$
$\Rightarrow x^2 - 37x - 8x + 296 = 0$
$\Rightarrow x(x - 37) - 8(x - 37) = 0$
$\Rightarrow (x - 37)(x - 8) = 0$
$\Rightarrow x - 37 = 0 or x - 8 = 0$
$\Rightarrow 3 = 37 or x = 8$
When x = 37, then
the length of pond = 50 - 2 × 37
$= 50 - 74$
$= -24m$
Length cannot be negative. So, x = 37 is rejected.
When x = 8, then
the length of pond = 50 - 2x
$= 50 - 2 × 8$
$= 50 - 16$
= 34m
and the breadth of the pond
$= 40 - 2x$
$= 40 - 2 × 8$
$= 40 - 16$
$= 24m$
Hence, the length and breadth of the pond are 34m abd 24m respectively.
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