Using Nonlinear programming

Problem 1. A forest fire is burning down a narrow valley of width 2 miles at a velocity of 32 feet per minute. A fire can be contained by cutting a fire break through the forest across the width of the valley. A man can clear 2 feet of fire break in a minute. It costs $20 to transport (includes return trip) each man to the scene of fire and each man is paid $6 per hour while there. The value of timber is $2000 per square mile. How many men should be sent to fight the fire so as to minimize the total costs? Problem 2. A perfume manufacturer wants to produce deodorant for both men and women. The sale price per gallon for Eau de Man is $3. Eau de Woman is $5 per gallon. Both use a discontinued mouth wash as the prime ingredient. One hundred gallons of the mouthwash have been provided free by a bankrupt competitor. Each gallon of Eau de Man uses 0.9 gallons of mouthwash while Eau de Woman uses 0.5 gallons. It costs $0.05 QW2 to produce QW gallons of Eau de Woman and $0.20(QM)3/2 to produce QM gallons of Eau de Man. The company wants to maximize their profit for disposing of the 100 gallons of mouthwash. (a) Formulate this problem in the standard mathematical programming framework. (b) Determine the optimal product mix using the Lagrange multiplier approach. Problem 3. Write down the Kuhn-Tucker conditions for the following problems (simplify and give the final form): (a) Minimize Z=X1X2X3+1/2(X12+X22+X32) Subject to X1+X2+X314X1+2X237X1,X2,X30