Question

A population gathers plants and animals for survival. They need at least 360 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 27 units of plants and 23 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.

Answer 1

Answer:

0 units of plants and 18 animals.

Explanation:

From the information provided, we know that this population must hunt at least 8 animals, to obtain the hides necessary to survive. When hunting such a quantity of animals, they would also obtain 160 units of energy and 200 units of protein. In terms of costs, this equals 80 hours of labor.

How can the population obtain the additional 200 units of energy and 100 of protein they need to live? They have two options: gather plants or keep hunting animals, of course, respecting the available quantities.

In the former case, at least 10 units of a plant would have to be gathered, this is the only way to obtain the additional units of energy and protein they are looking for. With 8 animals and 10 units of plants, they would have 460 units of energy, 300 of protein and 8 hides, and would have to invest 280 hours of labor.

If the population decided to reduce labor by gathering fewer plants (which is the activity that requires more time) but hunting the same amount of animals, it would obtain 430 units of energy, 290 units of protein and 8 hides. That means that it would not survive because it would not meet the requirement of 300 proteins.

Therefore, the only way to reduce hours of labor without risking survival would be gathering fewer plants but hunting more animals. Following this logic, the optimal amount would be 18 animals and 0 units of plants, as this would allow people to obtain the necessary resources to survive at the minimum cost in terms of labor. Furthermore, it would fit the physical limit of 27 units of plants and 23 animals. This combination would allow to obtain 360 units of energy, 450 of protein and 18 hides by investing only 180 hours of labor.

The attached image shows what happens when the quantity of animals is increased and that of plants is decreased until reaching the optimal combination of 18 animals and no units of plants.