A cereal company makes a cereal from two ingredients: wheat and oats. Each ingre

Question

A cereal company makes a cereal from two ingredients: wheat and oats. Each ingredient provides two essential nutrients: vitamin A and vitamin B. The company wants to know how many ounces of wheat and oats to include in each box of cereal so that the minimum requirements of 50 milligrams of vitamin A and 15 milligrams of vitamin B are satisfied. An ounce of wheat provides 10 milligrams of vitamin A and 2 milligrams of vitamin B. An ounce of oats provides 5 milligrams of vitamin A and 1 milligram of vitamin B. An ounce of wheat costs $0.03 and an ounce of oats costs $0.06. The company wants to minimize the total cost of production.

Explanation / Answer

let x ounces of wheat and y ounces of oats are used

then x ounces wheat gives=>10x miliigram Vitamin a and 2x milligram of vitamin b

y ounces of oats gives =>5y milligrams of vitamin a and ymilligrams of vitamin b

Total cost= .03x+.06y

10x+5y=50

2x+y=15

both equations cannot be true simulataneously

2x+y=10----------1

2x+y=15---------------2

let us take eqn 2 as eqn 2 produces more than sufficient amount of vitamin a but eqn 1 cannot produce suffiecient amount of vitamin b

so y=15-2x

cost c=.03x+.06(15-2x)

c=.03x+.9-.12x

c=.9-.09x

so the max value of c is obtained when x=10

but it cannot satisfy eqn 2

so let y=0=>x=7.5

which is possible and low cost case

so the mixture requires only 7.5 ounces of wheat

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