Question

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.

Answer 1

Given that a rectangular pond has to be constructed in the centre of a rectangular lawn of dimensions 50 m × 40 m

So, the distance between pond and lawn would be same around the pond.

Say x m.

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.