Linear Programming 1. Honors Alg 2/Trig A city in Nevada is planning a new power generating station. They can use either a nuclear power plant, a coal-fired plant, or both. You have been hired to optimize the numbers of megawatts (mw) of each kind of plant built in order to cmply with a new Envinronmental Protection Agency (EPA) rule which states that the cost of disposing of wastes must be a minimum. Let n be the number of mw generated by the nuclear plant, and let c be the number of mw generated by the coal plant. Let D be the number of dollars spent per day disposing of wastes. a) You find that it costs $60 per mw each day to dispose of nuclear wastes. It costs only $30 per mw each day to dispose of the ashes from the coal. Write an equation for the cost of waste disposal. Write inequalites expressing the following constraints: i) ii) iii) The maximum amount of nuclear power available is 24 mw. The two plants must be capable of generating at least 25 mw total. Nuclear fuel costs $100 per mw each day, and coal costs $200 per mw each day. The daily fuel cost cannot exceed $4200. Constructing a nuclear plant costs 5 million dollars per mw, and constructing coal-fired plants costs 4 million dollars per mw. The total cost of construction must not exceed 140 million dollars. For political reasons, the number of mw in the coal plant must be more than half the number of mw in the nuclear plant. b) iv) v) c) Plot the feasible region. d) Is it feasible to use no nuclear power? Justify your answer. e) Shade the portion of the feasible region in which the cost of disposal is at most $1200 a day. Find the optimum point, the point at which the company’s waste disposal cost is a minimum. What is this cost? f) 2. A city in southern Illinois gets its 100 million gallons per day (mgd) of water from wells, from purified river water, and from unpurified river water. r = amount of purified river water w = amount of water from wells u = amount of unpurified river water C = no. of dollars per day spent on getting water a. The cost of water from these 3 sources is: $60 per mgd for purified river water, $90 per mgd for well water, and $10 per mgd for unpurified river water. Write an equation expressing the cost. Write an equation expressing u in terms of r and w. Rewrite the equation in part (a) so that it is terms of only r and w. b. c. INEQUALITIES: -The total amount of purified river water and well water is no more than 100 mgd. -The concentrations of various impurities will be satisfactorily low if r and w obey the following: for calcium: w -4r + 80 , for phosphates: w (-7/5)r + 70 , for chlorides: w -0.5r + 45 , for sulfates: w (-1/3)r + 30 , and for silicates: w 22 . PLOT THE FEASIBLE REGION: a) Shade the portion of the feasible region in which the cost is at most $5000 per day. b) Find out how much of each type of water the city should obtain. What is the cost at this point?