Differentiation - Problems Involving Maximisation and Minimisation

Towns \(A\) and \(B\) are on opposite banks of a river \(\sqrt{5}\) km wide, but \(B\) is \(5\) km downstream from \(A\).

The Land Transport Department wants to build a bridge to connect the towns. However, it is calculated that the cost of building a bridge to building a road of the same length is \(3:2\).

It is decided that a bridge will be built from \(A\) to a point \(C\), \(x\) km from a point \(D\) directly opposite to \(A\) across the river and a road from \(C\) to \(B\).

Find the value of \(x\) to minimize the cost of building the bridge and the road.