The diagonal of a rectangular field is $15\text{m}$ and its area is $108{\text{m}}^2$. What is the cost of fencing the field at $\text{₹}50.50$ per $\text{m}$?

Let the length of the rectangular field be $l$ and the breadth be $b$.From the given properties:Diagonal, $\sqrt{l^2+b^2}=15⟹l^2+b^2=225$ Area, $l\times b=108$ We know the algebraic identity:
$$(l+b{)}^2=l^2+b^2+2lb$$
Substituting the values:
$$(l+b{)}^2=225+2(108)=225+216=441$$
$$l+b=\sqrt{441}=21\text{m}$$
The perimeter of the rectangular field (length of fencing needed) is:
$$\text{Perimeter}=2(l+b)=2\times 21=42\text{m}$$
The cost of fencing at $\text{₹}50.50$ per meter is:
$$\text{Total Cost}=42\times 50.50=42\times \frac{101}{2}=21\times 101=\text{₹}2121$$