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Assignment 8 Solutions (1206A8w2008)

# Assignment 8 Solutions (1206A8w2008) - Assignment#8 MGSC...

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MGSC 1206.2 Assignment #8 Winter 2008 TOTAL POINTS = 65 Problem 1: 17.3 #32. Suppose that the profit (in hundreds of dollars) of a certain firm is approximated by: P(x,y) = 1000 + 24x – x 2 + 80y – y 2 where x is the cost of a unit of labour and y is the cost of a unit of goods. Find values of x and y that maximize profit. Find the maximum profit. Solution: 1) x P x 2 24 - = , y P y 2 80 - = 2) = = = - = - = = 40 12 0 2 80 0 2 24 0 0 y x y x P P y x 3) 2 - = xx P , 2 - = yy P , 0 = xy P 4) 0 4 0 ) 2 ( 2 2 = - - - = - = xy yy xx P P P D & 0 2 < - = xx P . So ( x =12, y =40) gives the maximum profit. 5) 1600 3200 144 288 1000 40 40 80 12 12 24 1000 ) 40 , 12 ( 2 2 - + - + = - + - + = = = y x MaxP 744 , 2 \$ = Problem 2: 17.3 #34. The total cost (in dollars) to produce x units of electrical tape and y units of packing tape is given by: C(x,y) = 2x 2 + 3y 2 -2xy + 2x -126y + 3800 Find the number of units of each kind of tape that should be produced so that the total cost is a minimum. Find the minimum total cost. Solution: 1) 2 2 4 + - = y x C x , 126 2 6 - - = x y C y 2) 0 252 4 12 2 2 2 ... 0 126 2 6 1 ....... 0 2 2 4 0 0 = - - × = - - = + - = = x y e equation x y equation y x C C y x 12 0 48 4 0 2 25 2 4 : 1 25 0 250 10 1 2 2 = = - = + - = = - + × x x x e y y e e Critical point: ( x =12, y =25) 3) 4 = xx C , 6 = yy C , 2 - = xy C 4) 0 20 4 6 4 2 = - = - = xy yy xx C C C D & 0 4 = xx C . Hence ( x =12, y =25) gives the minimum cost.

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Assignment 8 Solutions (1206A8w2008) - Assignment#8 MGSC...

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