From: Clinton Priddle Newsgroups: sci.math.num-analysis Subject: Maths Assignment Date: Tue, 12 Aug 1997 21:54:31 +1000 A manufacturing company intends to make 2 types of plastics (Type A and Type B) from 2 base components - compound X and compound Y. The identities of compound X and Y are trade secrets. To make type A plastic 1.2Kgs of X and 0.8 kgs of Y are chemically combines to produce 1kg of type A plastic (ie 1.2X + 0.8Y = 1A) To make type B plastic 0.5kgs of X and 1.2kgs of Y are chemically combined to produce 1kg of type B plastic (ie 0.5X + 1.2Y = 1B) The profit on type A plastic is $100 per kg, and $150 per kg on type B plastic. (ie P=100A + 150B) The company can obtain Compounds X and Y in powder form by using 2 steel rectangular evaporation tanks over an eight hour period. To obtain compound X it takes 4 hours for 40% of the solvent in which compound X is dissolved to evaporate. This results in 40% of compound X precipitating out of the solutionwhich can then be collected by filtering (if 60% of solvent evaporates, then 60% of Compound X precipitates out). Since the change in volume is proportional to the amount of compound X present the following differential equation can be used to model the situation. dV/dt = kV Where V is the volume in cubic metres and t is the time in hours. The constant k can be deetermined from the above information. Once k is known the volume at any time (t) can be calculated and thus the coresponding percentage of compound X which has been precipitated can be calculated. The initial concentration of compound X is 3kg/cubic metre. Likewise it takes 2 hours for 30% of solvent containing compound Y to evaporate. This results in 30% of compound Y precipitating out of the solution which can be collected by filtering. The initial concentration of compound Y is 8kg/cubic metre. Finally, the manufacturing company must decide on the dimensions of the 2 steel evaporation tanks given that a 10X6 metre rectangular sheet of steel is available for tank construction. For ease of assembly the company decides to cut the steel sheet into two rectangular pieces (not necessarily equal) and remove a square section from each corner. The edges can then be folded up and welded into rectangular tanks. The volume of each tank depends on where the metal sheet is divided initially and also the size of the square section removed from each corner. For example, if the metal sheet is cut into two sections, 6X6 and 4X6 respectively. If the squares of side length x are removed from each corner the following rectangular tanks will result. (V = L X B X H) V = (6-2x)(6-2x) x V = (4-2x)(6-2x) x Questions 1. Determine the dimensions of the evaporation tanks which will result in the optimal production of Compounds X and Y. Also determine the amounts of Compound X and Y produced in an eight hour period. Present your findings in tabular form. 2. Determine the maximum amount of Type A and B plastic which could be produced in an eight hour period. Also determine the maximum profit. Present you findings in tabular form. --------------------------------------------------------------------- I hope you can help, and thank you for even looking at this for me. With the equation dV/dt = kV I can only work out a value of 'k' if I use a value for volume which is a percentage. Is that OK to do? I am very stumped with the shape of the tanks. If I am given the size of the 2 pieces the metal is cut into I can work out the value for the square cut out which will give the max volume. I cant work out which sizes to cut the metal into to start off with, however. I hope you can help, and thanks again for even looking at it. Clinton Priddle