Question

A livestock farmer has 500 acres to devote to grazing. He estimates that cattle require 5 acres per head and sheep require 3 acres per head. He has winter shelter facilities for 40 head of cattle and for 125 sheep. What constraints are imposed on the number of cattle and sheep he can raise?

Short answer

The constraints imposed on the number of cattle and sheep the farmer can raise are: 1. \(5x + 3y \le 500\) 2. \(x \le 40\) 3. \(y \le 125\) 4. \(x \ge 0\) 5. \(y \ge 0\)

Step-by-step solution

Land constraint

We know that the farmer has 500 acres of land to devote to grazing. Since a cattle requires 5 acres per head and a sheep requires 3 acres per head, we have the following constraint on the number of cattle and sheep: \(5x + 3y \le 500\)

Winter shelter constraint for cattle

The farmer has winter shelter facilities for maximum 40 heads of cattle. Therefore, the following constraint is imposed on the number of cattle: \(x \le 40\)

Winter shelter constraint for sheep

Similarly, the farmer has winter shelter facilities for maximum 125 heads of sheep. This imposes the following constraint on the number of sheep: \(y \le 125\)

Non-negative constraint

Since the number of cattle and sheep cannot be negative, we also have the following non-negative constraints: \(x \ge 0\) \(y \ge 0\) So, the constraints imposed on the number of cattle and sheep the farmer can raise are: 1. \(5x + 3y \le 500\) 2. \(x \le 40\) 3. \(y \le 125\) 4. \(x \ge 0\) 5. \(y \ge 0\)

Key concepts

Land Resource Constraints

When it comes to agriculture, especially livestock farming, land resource constraints play a crucial role in determining the capacity and productivity of the farm. The amount of land available for activities such as grazing directly influences how many animals a farm can support. In our sample exercise, the farmer has 500 acres of land, which limits the number of cattle and sheep they can keep. This is represented mathematically by the inequality equation \(5x + 3y \le 500\) where \(x\) and \(y\) represent the number of cattle and sheep, respectively.

The constraint takes into account the acreage requirements per head of livestock—5 acres for each head of cattle and 3 acres for each sheep. These requirements are based on factors such as the forage availability and the nutrition needs of the animals. Farmers must balance the land they have with these needs to avoid overgrazing, which can lead to land degradation and reduced productivity over time.

Livestock Management

Livestock management is a key aspect of agricultural planning that includes considerations like feeding, breeding, and providing shelter for the animals. One of the main concerns highlighted in our exercise is the availability of winter shelter, which can be a limiting factor for livestock numbers. In our farmer's case, shelter capacity is restricted to 40 heads of cattle and 125 heads of sheep, leading to the constraints \(x \le 40\) and \(y \le 125\).

This represents the maximum number of animals that can be protected from harsh weather conditions, an essential factor for their survival and welfare. Efficient livestock management requires planning for all seasons, ensuring that there is adequate infrastructure to support the animals' needs throughout the year. It is also important to consider local climate, potential diseases, and overall farm management practices when determining these constraints.

Inequality Constraints

Inequality constraints are essential components of linear programming models, such as the one presented in our exercise for livestock farming. They form the basic structure that defines the limits within which the farmer must operate. The constraints \(5x + 3y \le 500\), \(x \le 40\), and \(y \le 125\) tell us the maximum allowable limits, while the non-negative constraints \(x \ge 0\) and \(y \ge 0\) ensure that the solution makes sense in real-world terms, as the number of animals cannot be negative.

Inequality constraints reflect real-world limitations like resource availability and capacity restrictions. They help in defining the feasible region of operation and in finding the optimal solution that maximizes or minimizes the objective, such as profit or crop yield, while staying within these defined boundaries. Understanding and applying these constraints correctly is vital for efficient and sustainable farm management.