Question

An anthropology article presents a hypothetical situation that could be described by a linear programming model. Suppose a population gathers plants and animals for survival. They need at least 360 units of energy, 300 units of protein, and 8 hides during some time period. One unit of plants provides 30 units of energy, 10 units of protein, and no hides. One animal provides 20 units of energy, 25 units of protein, and 1 hide. Only 25 units of plants and 25 animals are available. It costs the population 30 hours of labor to gather one unit of a plant and 15 hours for an animal. Find how many units of plants and how many animals should be gathered to meet the requirements with a minimum number of hours of labor.

Answer 1

Answer:

The minimum  labor = 270 hours and it is obtained when 0 units of plants and 18 animals are collected.

Step-by-step explanation:

We assume that x should represent the number of units of plants

Also y to represent the number of animals.

So, minimze (z) = 30 x + 15 y

Subject to:

30 x + 20 y ≥  360  

10  x + 25 y ≥ 300

               y  ≤ 8

 0  ≤  x  ≤ 25

 0  ≤  y  ≤ 25

Changing the above linear inequalities into linear equation; we have:

30 x + 20 y =  360  

10  x + 25 y = 300

               y  = 8

The graph for the above discussion can be located in the attached diagrammatic expression below:

NOW, the corner points (0,18),(0,25),(25,25), and (25,8) can be calculated from the graph. From the diagram, the corner point  $[\frac{60}{11} , \frac{108}{11} ]$ can be found by solving the system.

30 x + 20 y =  360  

10  x + 25 y = 300

Corner point (10, 8) = 10 x + 25 y = 300

                                                   y  = 8

Corner point                               Value of z = 30 x + 15 y

(0,18)                                            270

(0,25)                                           375

(25,25)                                         1125

(25,8)                                            870

$[\frac{60}{11} , \frac{108}{11} ]$                                          $\frac{3420}{11}$ ≅ 310.91

(10, 8)                                            420

∴

The minimum  labor = 270 hours and it is obtained when 0 units of plants and 18 animals are collected.