The cost of fencing of an equilateral triangular park and a square park is the same. If the area of the triangular park is $16\sqrt{3}$ m$^2$, then the length of the diagonal of the square park is :

The cost of fencing an equilateral triangular park and a square park is the same, implying the perimeter of both parts is equal.

Let $a$ be the side of the equilateral triangular park and $s$ be the side of the square park.

Thus, $3a = 4s \rightarrow s = \dfrac{3}{4}a$

Now, area of triangular park : $\dfrac{\sqrt3}{4} a^2 = 16 \sqrt3$

Hence, $a^2 = 64$

$a = 8$ units (since side of an equilateral triangle can't be negative)

Thus, $s = \dfrac{3}{4} \times 8 = 6$ units.

Now, length of diagonal $= \sqrt 2 \times s = \sqrt2 \times 6 = 6\sqrt2$ units

Hence, the correct option is Option 4.