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Unformatted text preview: Ma 116  Calculus II Hw6 (Solutions) Due: Apr 25, 2011 Pledge and Sign : An important component of these written exercises is the quality of your presentation, including: legibility, organization of the solution, and clearly stated reasoning where appropriate. Points will be deducted for sloppy work or insufficient explanations. 1. A firm manufacturers a commodity at two different locations. If q 1 and q 2 represent the monthly output from each factory, then the monthly cost of production is approximated by the function, C = f ( q 1 ,q 2 ) = 2 q 2 1 + q 2 2 + q 1 q 2 + 500 . If the company needs to deliver 200 units per month, how many units should be supplied by each factory in order to minimize the production costs? Solution : Maximize f ( q 1 ,q 2 ) subject to the constraint q 1 + q 2 = 200. Method1 : Use the constraint, q 2 = 200 q 1 , to reduce the problem to optimization in one variable. Maximize C = h ( q 1 ) = 2 q 2 1 + (200 q 1 ) 2 + q 1 (200 q 1 ) + 500, on the interval 0 q 1 200. h ( q 1 ) = 0 4 q 1 200 = 0 cost is minimized for q 1 = 200 / 4 = 50 and q 2 = 150. Method2 : Use Lagrange Multipliers and solve the unconstrained optimization for L ( q 1 ,q 2 , ) = f ( q 1 ,q 2 ) ( q 1 + q 2 200). Setting L = ~ yields a system of three equations: 4 q 1 + q 2 = ; 2 q 2 + q 1 = ; q 1 + q 2 200 = 0 ....
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This note was uploaded on 10/09/2011 for the course MATHEMATIC MA116 taught by Professor Duboski during the Fall '11 term at Stevens.
 Fall '11
 Duboski
 Calculus

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