The customers of a particular company are located throughout an area comprised of S square miles, and they are serviced from k warehouses. On average, the distance in miles between a warehouse and a customer is given by the formula (S/k)0.5. The annual capital cost of building a warehouse is $40,000 and the annual operating cost of running a warehouse is $50,000. Annual shipping costs average $1 per mile per customer.
Suppose that the current market size is 250,000 customers, spread out over an area of 600 square miles. What is the optimal number of warehouses for the firm to operate?