OPTIMIZATION ASSIGNMENT 1. An employee's monthly production M. in number of units produced, is found to be a func- tion of the number of year of service, t. For a certain product, a productivity function is being given by: M(t) = -2+ +100t+180, ost s 40 Find the maximum productivity and the year in which it is achieved. 2. A firm determines that its total profit in dollars from the production and sale of x units of a product is given by: P(x) = 1500 x 2 0 x2 - 6x + 10 Find the number of units x for which the total profit is a maximum. Technicians working for the Ministry of Natural Resources have found that the amount of a pollutant in a certain river can be represented by where t is the time (in years) since a clean up campaign started. At what time was the pollution at its lowest level? P(5) = 2t + Osts1 162t + 1 4. A farmer has 54 m of fencing with which to build two animal pens with a common side. One pen is rectangular; the other is square. If the area of the pens is to be maximized, what are their dimensions? 5. Find the length of the sides of the isosceles triangle of greatest area that has a perimeter of 18 cm. 6. Find the point on the parabola x = y2 - 8y + 18 closest to (-2,4). 7. A light airplane flying east at 300 km/h passes over Buttonville airport 10 min before a sec- ond airplane flying southeast 420 km/h passes over the same point. Assuming that both the airplanes are at the same altitude, what is the minimum distance between the planes? 8. A rectangular field along a straight hedge is to be divided into five smaller fields by one fence parallel to the hedge and five fences perpendicular to the hedge. Find the maximum area that can be enclosed if 1600 m of fencing is available. 9. A box with an open top is to be made from a square piece of cardboard, of side length 200 cm, by cutting a square from each corner and then folding up the sides. Find the dimensions of the box of largest volume.[PLEASE CHOOSE FOUR OF THE NINE QUESTIONS ASSIGNED TO SOLVE AND SUBMIT FOR GRADING. YOU MAY ONLY CHOOSE TWO OUESTIONS FROM #'5 1 - 3.)