There are 100 inhabitants in the village A and 50 inhabitants in the village B. In what point of the road connecting these two villages should be a bathhouse built in order to minimize the total distance passed by all 150 inhabitants to the bathhouse and back?
Let the distance between villages be a. Suppose that the bathhouse is built at a distance x from A (0 ≤ x ≤ a). Then the total distance will be 2(100x + 50(a − x)) = 200x + 100a − 100x = 100x + 100a. It is minimal when x = 0. This means that the bathhouse should be built in the village A.