3. Janet Johnson has been watching the Dr. Iz show and concludes that her family

Question

3. Janet Johnson has been watching the Dr. Iz show and concludes that her family needs at least 50 grams of protein and 15 milligrams of iron per day to maintain good health. Since Janet has recently converted to a vegan diet, she wants to make sure that the family gets all these nutrients solely from lentils and spinach. Each cup of lentils costs an average of $0.15 and contains 18 grams of protein and 5.5 milligrams of iron. Each cup of spinach costs $0.50 and has 1.1 grams of protein and 1.3 milligrams of iron.

Write the linear programming formulation to help Janet determine the quantities of lentils and spinach to meet the nutritional requirements at least cost.

Explanation / Answer

Suppose Janet Johnson for her family needs x cups of lentils and y cups of spinach

To minimize the cost

Min Z = 0.15 x + 0.5 y

To get at least 50 grams of protein and 15 milligrams of iron per day to maintain good health we have following constrains

18 x + 1.1 y 50

5.5 x + 1.3 y 15

Where x and y 0

Now to get the values of x and y we have to convert the above constrains in following equations

18 x + 1.1 y = 50 …………………………………. (1)

5.5 x + 1.3 y = 15 …………………………………… (2)

By Multiplying eq (1) by 1.3 and eq (2) by 1.1, we get

23.4 x + 1.43 y = 65 ………………………………………………… (3)

6.05 x + 1.43 y = 16.5 ………………………………………………. (4)

Now subtracting eq (4) from eq (3) we get

17.35 x = 48.5 or x = 2.80

Now putting the value of x in equation 1 or 2 we are getting negative value of y which is not acceptable as we know that y 0 so assume that y =0

Janet Johnson needs at list x = 2.78 to get the nutritional required amount of protein and iron.

And cost Z = 0.15 * 2.78 = $ 0.42

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