The width of the path around a square field is 4.5 m and its area is 105.75 m². Find the cost of fencing the path at the rate of Rs. 100 per metre.

Let the length of interior square = $$x$$
The length of exterior square = ($$x$$ + 9)
According to question
$$\eqalign{ & {\left( {x + 9} \right)^2} - {x^2} = 105.75 \cr & \left( {x + 9 + x} \right)\left( {x + 9 - x} \right) = \frac{{10575}}{{100}} \cr & \left( {2x + 9} \right) \times 9 = \frac{{423}}{4} \cr & 2x + 9 = \frac{{47}}{4} \cr & 2x = \frac{{47}}{4} - 9 \cr & 2x = \frac{{11}}{4} \cr & x = \frac{{11}}{8} \cr & {\text{Total cost of fencing}} \cr & = \frac{{11}}{8} \times 4 \times 100 \cr & = {\text{Rs}}{\text{. }}550 \cr} $$