A population gathers plants and animals for survival. They need at least 420 uni

Question

A population gathers plants and animals for survival. They need at least 420 units of energy, 300 units of protein, and 8 hides. One unit of plants provides 30 units of energy, 10 units of protein, and 0 hides. One animal provides 20 units of energy, 25 units of protein, and 1 hide. Only 22 units of plants and 30 animals are available. It costs 20 hours of labor to gather one unit of a plant and 10 hours for an animal. Find how many units of plants and animals should be gathered to meet the requirements with a minimum number of hours of labor.

Explanation / Answer

Let x represent the number of plants gathered and y the number of animals.

The objective function is C=20x+10y (20 hours per unit of plants, and 10 hours per unit of animal) -- we wish to minimize the objective function.

We are subject to the following constraints:

[30x+20y>=420] or [3x+2y>=42] (The amount of energy taken in must be at least 420 units.)

[10x+25y>=300] or [2x+5y>=60] (The amount of protein must be at least 300 units.)

[0x+y>=8] or [y>=8] (The number of hides must be at least 8 -- so the number of animal units must be at least 8.)

[0<=x<=22] (The natural constraint of no negative plant units allowed, and the maximum number of plant units available is 22.)

[0<=y<=30] The number of animal units available is 30.

Here is a graph of the feasible region:

There are 6 points of intersection bounding the feasible region. They are (0,21),(0,30),(22,30),(22,8),(10,8), [(90/11,96/11)] .

C(0,21)=210

C(0,30)=300

C(22,30)=740

C(22,8)=520

C(10,8)=280

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