Question

In a rectangular Park of dimensions 50 m * 40 m a rectangular pond is constructed so that the area of grass strip of uniform with surrounding the point would be 1184 m square find the length and breadth of the pond

Answer 1

The length of the pond is 39.75 m and the width is 29.75 m.

Explanation:
The width would be given by (40-x), where x is the width of the grass strip. The length would be (50-x). We know that their area is 1184; the area is found by multiplying the width and length:

(40-x)(50-x)=1184.

Multiplying these binomials, we have
40*50-40*x-50*x-x*(-x)=1184
2000-40x-50x+x²=1184

Combining like terms, we have
2000-90x+x²=1184.

We subtract 1184 from both sides:
2000-90x+x²-1184 = 1184-1184
816-90x+x²=0.

Use the quadratic formula to solve this:
$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \\ \\x=\frac{--90\pm \sqrt{(-90)^2-4(1)(816)}}{2(1)} \\ \\x=\frac{90\pm \sqrt{8100-3264}}{2}=\frac{90\pm \sqrt{4836}}{2} \\ \\x=\frac{90\pm 69.5}{2}=\frac{90+69.5}{2}\text{ or }\frac{90-69.5}{2} \\ \\x=\frac{159.5}{2}\text{ or }\frac{20.5}{2}=79.75\text{ or }10.25$

Since the entire length of the rectangle is only 50 m, the width of the strip around the pond cannot be 79.75; this means the width of the grass strip must be 10.25. This makes the length of the pond 50-10.25=39.75 and the width of the pond 40-10.25=29.75.

Answer 2

     see the attached picture to better understand the problem

we have that
dimensions of rectangular park = 50 m x 40 m
So
 Area of park = 50*40 = 2000 m²
Area of the grass strip surrounding the pond = 1184 m²
Hence,
area of the pond = 2000-1184 = 816 m²                   
Let
x--------> the width of the grass strip 
So,
 (50-2x)(40-2x) = 816
⇒ 2000 - 100x - 80x + 4x² = 816
⇒ 2000-180x+4x² = 816⇒4x²-180x+1184 = 0
⇒x²-45x+296 = 0
⇒x²-37x-8x+296=0

using a graph tool
see the attached figure N 2

the solution is 
x=8
Width of the grass strip = 8 m

Length of the pond = 50-16 = 34 m
Breadth of the pond = 40-16 = 24 m