A company is developing a new additive for gasoline. The additive is a mixture o

Question

A company is developing a new additive for gasoline. The additive is a mixture of three liquid ingredients,, and For proper performance, the total amount of additive must be at least 11 oz per gal of gasoline. However, for safety reasons, the amount of additive should not exceed 17 oz per gal of gasoline. At least oz of ingredient I must be used for every ounce of ingredient I, and at least 1 oz of ingredient IlI must be used for every ounce of ingredient I. The costs of dlIl are $0.30, 50.12, and $0.42 per oz, respectively. (a) Find the mixture of the three ingredients that produces the minimum cost of the additive Dozofingredient ll, andOozof The minimum cost mixture has oz of ingredient I ingredient l (Type an integer or a decimal. Round to three decimal places as needed.) (b) How much of the additive should be used per gallon of gasoline? oz of additive should be used per gallon of gasoline. (Type an integer or a decimal. Round to three decimal places as needed.)

Explanation / Answer

Let the number of oz of ingredient 1 be x1

Let the number of oz of ingredient 2 be x2

Let the number of oz of ingredient 3 be x3

Cost Function: (Minimize) z = 0.30x1 + 0.12x2 + 0.42x3

Constraints:

x1 + x2 + x3 >= 11

x1 + x2 + x3 <= 17

x1 >= x2/4

x3 >= x1

Now solving the constraints,

So for minimizing the cost we can keep x2=4x1 and x1=x3

4x1 + x1 + x1 = 11

x1 = 11/6

x3 = 11/6

x2 = 44/6 = 22/3

Hence the answer will be 11/6,22/3,11/6

minimum cost = 2.2$

b) Hence the answer is 11/6 + 22/3 + 11/6 = 11

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