Question

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

A. Length $ = 34m$, Breadth $ = 24m$
B. Length $ = 24m$, Breadth $ = 34m$
C. Length $ = 37m$, Breadth $ = 8m$
D. None of these

Answer 1

Hint: For solving this, consider the distance between the boundaries of the lawn and the pond be $x$. Then the length of the pond is equal to the length of the lawn minus twice the distance between the boundaries and the breadth of the pond is equal to breadth of the lawn minus twice the distance between the boundaries.

Complete step by step solution:
It is given in the question that,
Length of the lawn is $50$m,
Breadth of the lawn is $40$m,
And the area of the grass surrounding the pond is $1184sq.m$.
Let us consider the distance between the boundaries of the lawn and the pond to be $x$ .
Therefore, we get ,
The length of the pond is equal to the length of the lawn minus twice the distance between the boundaries.
Mathematically we can write this as,
Length of the pond $ = (50 - 2x)$ m ,
Similarly, we can say that the breadth of the pond is equal to the breadth of the lawn minus twice of the distance between the boundaries.
Mathematically we can write this as,
Breadth of the pond $ = (40 - 2x)$ m ,
According to the question,
Area of the pond is given as area of the lawn minus the area of the grass around pond ,
Mathematically, we can write ,
Area of the pond $ = $ Area of the lawn $ - $ Area of the grass around that pond.
It is given as ,
$ \Rightarrow (40 - 2x)(50 - 2x) = (50)(40) - 1184$
Now simplify the above expression,
$ \Rightarrow 4{x^2} - 180x + 2000 = 2000 - 1184$
$
   \Rightarrow {x^2} - 45x + 296 = 0 \\
   \Rightarrow x(x - 8) - 37(x - 8) = 0 \\
   \Rightarrow (x - 8)(x - 37) = 0 \\
  x = 37,x = 8 \\
 $
If $x = 8,l = 34m,b = 24m.$
$x = 37$ gives unrealistic values. Therefore, rejecting this value.

Therefore, we can say that option ‘A’ is the correct option.

Note: Questions similar in nature as that of above can be approached in a similar manner and we can solve it easily. The important thing to recollect about any equation is that the ‘equals’ sign represents a balance.