Refer to Example 9. How many packets of each food should be used to maximize the amount of vitamin A in the diet? What is the maximum amount of vitamin A in the diet?
Let the number of packets of food P be x and the number of packets of food Q be y .
We have to tabulate the given data as,
Resources FoodsP FoodsQ Requirement Calcium 12 units 3 units At least 240 units Iron 4 units 20 units At least 460 units Cholesterol 6 units 4 units At most 300 units Vitamin A 6 units 3 units
The equation for calcium is given as,
12 x + 3 y ≥ 240 4 x + y ≥ 80
The equation for iron is given as,
4 x + 20 y ≥ 460 x + 5 y ≥ 115
The equation for cholesterol is given as,
6 x + 4 y ≤ 300 3 x + 2 y ≤ 150
We need to maximize the amount of vitamin A so we can use function which will Maximize Z .
Maximize Z = 6 x + 3 y (1)
All constraints are given as,
4 x + y ≥ 80 x + 5 y ≥ 115 3 x + 2 y ≤ 150 x ≥ 0 , y ≥ 0
4 x + y ≥ 80
x 0 20 y 80 0
x + 5 y ≥ 115
x 0 70 y 23 9
3 x + 2 y ≤ 150
x 0 50 y 75 0
Plot the graph using equations of constraint.
Substitute the value of x and y in the equation (1) to check the max value.
Corner points Value of Z ( 2 , 72 ) 228 ( 15 , 20 ) 150 ( 40 , 15 ) 285 (Maximum)
Thus, the amount of vitamin A will be maximum if 40 packets of food P and 15 packets of food Q are used and the maximum amount of vitamin A is 285 units.
Let the number of packets of food P be
We have to tabulate the given data as,
| Resources | FoodsP | FoodsQ | Requirement |
| Calcium | | | At least |
| Iron | | | At least |
| Cholesterol | | | At most |
| Vitamin A | | |
The equation for calcium is given as,
The equation for iron is given as,
The equation for cholesterol is given as,
We need to maximize the amount of vitamin A so we can use function which will
All constraints are given as,
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
Plot the graph using equations of constraint.
Substitute the value of
| Corner points | Value of Z |
| | |
| | |
| | |
Thus, the amount of vitamin A will be maximum if
