Question

Bob, a nutritionist who works for the University Medical Center, has been asked to prepare special diets for two patients, Susan and Tom. Bob has decided that Susan's meals should contain at least 490 mg of calcium, 23 mg of iron, and 60 mg of vitamin C, whereas Tom's meals should contain at least 440 mg of calcium, 18 mg of iron, and 50 mg of vitamin C. Bob has also decided that the meals are to be prepared from three basic foods: Food A, Food B, and Food C. The special nutritional contents of these foods are summarized in the accompanying table. Find how many ounces of each type of food should be used in a meal so that the minimum requirements of calcium, iron, and vitamin C are met for each patient's meals.

Contents (mg/oz)
Calcium Iron Vitamin C
Food A 30 1 2
Food B 25 1 5
Food C 20 2 4

Susan's meals:

Food A     oz
Food B     oz
Food C     oz

Tom's meals:

Food A     oz
Food B     oz
Food C     oz

Answer 1

Let Susan’s meal consist of x ,y and z oz of Food, A,B,C respectively and let Tom’s meal consist of a ,b and c oz of Food, A,B,C respectively.

Then 30x+25y +20z ≥ 490 or, on dividing both the sides by 5, 6x+5y+4z ≥ 98…(1), x+y+2z ≥ 23 …(2) and 2x+5y +4z ≥ 60…(3)

Also, 30a+25b +20c ≥ 440 or, on dividing both the sides by 5, 6a+5b+4c ≥ 88 …(4), a+b+2c ≥ 18 …(5) and 2a+5b +4c ≥ 50…(6)

Now, on multiplying both the sides of the 2^nd inequality by 2 and on subtracting the result from the 1^st inequality, we get 6x+5y+4z -2x-2y-4z ≥ 98-46 or, 4x+3y ≥ 52 or,                                   y ≥ -(4/3)x+52/3 …(7).

Similarly, on multiplying both the sides of the 2^nd inequality by 2 and on subtracting the result from the 3^rd inequality, we get 2x+5y+4z-2x-2y-4z ≥ 60-46 or3y ≥ 14 or, y ≥ 14/3…(8).

A graph of the lines y = -(4/3)x+52/3 and y = 14/3 is attached. The feasible region is on and above these 2 lines in the 1^st quadrant of the xy plane. These 2 lines meet at the point (9.500,4.667) ( on rounding off to 3 decimal places) so that the least possible values of x and y are 9.500 and 4.667 respectively. Further, on substituting x = 9.500 and y = 4.667 in the 2^nd inequality, we get 9.500+4.667 +2z ≥23 or, 2z ≥ 8.833 so that z ≥ 4.417 ( on rounding off to 3 decimal places).

Thus, Susan's meals are as under:

Food A     9.500 oz
Food B    4.667 oz
Food C     4.417 oz

On multiplying both the sides of the 5^th inequality by 2 and on subtracting the result from the 4^th inequality, we get 6a+5b+4c -2a-2b-4c ≥ 88-36 or, 4a+3y ≥ 52 or,                                   b ≥ -(4/3)a+52/3 …(9).

Similarly, on multiplying both the sides of the 5^th inequality by 2 and on subtracting the result from the 6^th inequality, we get 2a+5b+4c-2a-2b-4c ≥ 50-36 or3b ≥ 14 or,                       b ≥ 14/3…(10).

The graph of the lines b = -(4/3)a+52/3 and b= 14/3 is same as the earlier graph so that the least possible values of a and b are 9.500 and 4.667 respectively. Further, on substituting a = 9.500 and b = 4.667 in the 5^th inequality, we get 9.500+4.667 +2c ≥ 18 or, 2c ≥ 3.833 so that c ≥ 1.917( on rounding off to 3 decimal places).

Thus, Tom's meals are as under:

Food A     9.500 oz
Food B    4.667 oz
Food C     1.917 oz