Chapter 11, Problem 19RQ

As an engineer, suppose you are directed to design a pumping system to safely discharge a toxic industrial waste into a municipal reservoir. The concentrated wastewater from the plant will be mixed with freshwater from the lake, and this mixture is to be pumped into the center of the lake. You realize that the more water you mix with the waste, the more dilute it will be and thus the smaller the impact on the fish in the lake. On the other hand, the more water you use, the more it costs in electricity for pumping. Your objective is to determine the optimum amount of water to pump so the overall cost is a minimum.

• Assume that the cost of pumping is given by the expression C_pump = 10 Q?. The cost C_pump [$/day] depends on the pumping rate Q [gallons per minute, or gpm] of the water used to dilute the industrial waste.
• Now, suppose that some biologists have found that as more and more water dilutes the waste, the fish loss C_fish [$/day] can be expressed as C_fish = 2250 – 150 Q.

With this information, construct a graph, with pumping rate on the abscissa showing the pumping cost, the fish-loss cost, and total cost on the ordinate. For the scale, plot 0 to 15 gallons per minute for flow rate.

Determine both the minimum cost and the corresponding flow rate. Indicate the location of this answer on your graph.