Explanation:

- First, determine the area of the path. The given area of the path is 105.75 m².

- The path surrounds a square field and has a width of 4.5 m on all sides.

- Let the side of the square field be "a" meters.

- The length of the side of the larger square (field + path) is $a+2\times 4.5$. Therefore, it is $a+9$ meters.

- The area of the larger square (field + path) is $(a+9{)}^2$.

- The area of the field is $a^2$.

- The path's area is given by $(a+9{)}^2-a^2=105.75$.

1. Solve $(a+9{)}^2-a^2=105.75$ to find "a".

2. Simplifying, you find $18a+81=105.75$, resulting in $a=1.375$ meters.

3. Calculate the perimeter of the larger square $(4\times (a+9))$.

4. Cost of fencing is $(4\times 10.375)\times 100$.

The correct answer is Rs600.

- Therefore, the correct option is:

- Option 2: Rs600