A dietician is planning a snack package of fruit and nuts Ea

A dietician is planning a snack package of fruit and nuts. Each ounce of fruit will supply zero units of protein, 2 units of carbohydrates, and 1 unit of fat, and will contain 21 calories. Each ounce of nuts will supply 3 units of protein, 1 unit of carbohydrate, and 2 units of fat, and will contain 31 calories. Every package must provide at least 9 units of protein, at least 10 units of carbohydrates, and no more than 11 units of fat. What is the least number of calories possible in a package?

Solution

let \'x\' be the amount of fruit in ounces

let \'y\' be the amount of nuts in ounces

then

0.x +3y >= 9

2x+1y >=10

1x+2y <= 11

x>=0 and y>=0

let z be the number of calories , then our objective function is z= 21x + 10y which is to be minimized

3y =9

2x+y =10

y =3

2x = 7

x =7/2

point isA(7/2 , 3)

slove 3y =9

x+2y =11

x+6 =11

x =5

point isB(5,6)

solve

2x+1y =10

1x+2y = 11

3x = 9

x =3

6 +y=10

y =4

point is C(3,4)

evaluate z at A,B,C

Z at A = 21 *7/2 + 10*3 = 73.5 +30 =103.5

Z at B = 21*5 + 10*6 = 165

Z ar C = 21*3 +10*4 = 63 +40 =103

least number os calories possible is 103 and 3 ounce fruits and 4 nuts