Question

In Jen’s garden shop she makes two kinds of mixtures for planting: Gardening Mixture and Potting mixture. A package of Gardening mixture requires 2lb of soil, 1lb of peat moss, and 1lb of fertilizer. A package of potting mixture requires 1lb of soil, 2lb of peat moss, and 3lb of fertilizer. She has at most 64lb of soil, 44lb of peat moss, and 60lb of fertilizer. A package of garden mixture sell for $9.50 and a package of potting mixture sells for $2.50. Assuming all the mixtures she made will sell, how many packages of each type mixture should she make to maximize profit? Keep in mind that Jen has to produce at least 2 of each type of mixture.

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Answer 1

Answer:

	Soil	Peat Moss	Fertilizer
Gardening Mixture	2	1	1
Potting Mixture	1	2	3
Capacity	64	44	60
Assume,
x = no. of gardening mixture package
y = no. of potting mixture package
Objective function Max z = 9.5x + 2.5y
Subject to, C1	2x + y ≤ 64	x	0	32
				y	64	0
	C2	x + 2y ≤ 44	x	0	44
				y	22	0
	C3	x + 3y ≤ 60	x	0	60
				y	20	0
	C4	x  ≥ 2
	C5	y ≥ 2

C1 50 50 40 C2 30 C3 (2, 19.33) (12,16) 10 (28,8) C5 (2,2) -30 -20 -10 O 40 50 60 70 80 90 (31,2) C4

Coordinates of the feasible region or the optimal solutions are: (2,19.33) , (12,16) , (28,8) , (31,2) and (2,2)
For optimal solution, putting the coordinates in objective function
Max z = 9.5x + 2.5y
z(2,19.33) = 9.5(2) + 2.5(19.33) = 67.325
z(12,16) = 9.5(12) + 2.5(16) = 154
z(28,8) = 9.5(28) + 2.5(8) = 286
z(31,2) = 9.5(31) + 2.5(2) = 299.5
z(2,2) = 9.5(2) + 2.5(2) = 24
So, the optimal no. of packages of garden mixture and potting mixture is 31 and 2 respectively giving the maximum profit.