4. On the bank of the river is the location $A$, and downstream from it, further away from the bank, is the location $B$. Determine where the port $C$ should be built so that the transportation from $A$ to $B$, via $C$, is the cheapest, given that the cost of river transportation is half that of land transportation. It is assumed that the flow of the river and the path from $C$ to $B$ are straight lines (diagram on the right).
Solution. Let $C$ be the landing point. Let $l$ be a ray starting from $A$ on the side of the line $A C$ (the river) where point $B$ is not, which forms a $30^{\circ}$ angle with the line $A C$, and let $D$ be the foot of the perpendicular dropped from point $C$ to the ray $l$ (see the diagram on the right). Then $A C=2 C D$. The cost of transporting the goods is proportional to

$$
A C+2 C B=2 C D+2 C B=2(C D+C B)
$$

![](https://cdn.mathpix.com/cropped/2024_06_05_72b79cc38cc70f646757g-02.jpg?height=810&width=472&top_left_y=956&top_left_x=992)
which means it is also proportional to $C D+C B$, and will be minimized if points $B, C, D$ are collinear, i.e., if the line $C B$ forms a $60^{\circ}$ angle with the downstream direction of the river.

## Second Year