Question

LP6. A company manufactures two types of cleaners. One is an all-purpose cleaner (AP) and the other is an industrial strength cleaner (iS). Each cleaner is a mixture of three chemicals, as shown below. Ap e S All-purpose, AP Each kiloliter requires: 12 liters of surfactants 9 liters of enzymes 30 liters of solvents Industrial strength, IS Each kiloliter requires: 24 liters of surfactants 5 liters of enyzmes 30 liters of solvents The profit per kiloliter from the AP cleaner is $100 and the profit per kiloliter from the lIP cleaner is $85. The inventory of the company shows 480 liters of surfactants, 180 liters of enzymes, and 720 liters of solvents available. Assuming the company can sell allte cleaner it produces, find the number of kiloliters of each cleanser the company should produce to maximize profit. What is the maximum profit?

Be sure to show the graph of the feasible region, the appropriate vertices, and optimal values

Answer 1

Answeri- Lipp Graphic methodi- Given information: I cleaners All pupose Industrial Availala Surfactants 12 au 480 enfymes a 180 Solvent 30. 30. L 730 profits => $ 100x + $ 85 *> maximizing 2 = 100x + 85 ya Sub tot 1axt Quy 480 taxt 5 y 180 30x + 304 & 720 - x, y > 0 * constraints:- axt quy = 480 x=0] 1210) + guy = 480 y=[480/au ] y = [20] 90 12x tau (o)=480 12x=480 x=[u80/12 x= [40]

x 0 20 | 360 Ⓡ 9x + 5y = 180 1x=07 9 (o)t sy = 180 4-180150 y = [36 4:0] Tax +50)=180 x=180/9 x= [20 ] 3ox + 30y = 720 x ogu 1 x 0 30100+ 3oy = 720 T y 124 y = [720/30 ] y=[au / y=0 3ox + 30(0)=720 30x=720 Tx=720/30 x=aul * plotting the graph:- do to f f F to y a del ob u 8 12 16 20 24 28 32 36 40

* feasible region. a, b, c, d, e Corner 20 15 8 T * obiective function', Z = 100x+854 [maximizing a z=100 (6) + 20885 C 1700 b] 7=looco) + 85 co = 0 c] /= 100 cao) + 85co 2000 d] Z=100c15) + 85 (9) 2265 e] t = 100(8) + 85 (16) = 2160 Optimal value = corhee point (d). when x = 15 & 2265 at the & £ is maximized y = 9.

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