To create a mathematical model for finding the optimal daily diet for animals based on the provided table, we can set up a linear programming (LP) problem. Here’s a breakdown of the model:

Variables

Let:

• $x$ represent the amount (in sh.b.) of the first type of feed.
• $y$ represent the amount (in sh.b.) of the second type of feed.

Objective Function

The goal is to minimize the cost of the feed while meeting the nutritional requirements. The cost per unit is:

• $3x$ for the first type.
• $5y$ for the second type.

Thus, the objective function to minimize the cost is: $\text{Minimize } Z = 3x + 5y$

Constraints

Based on the nutritional requirements provided:

1. Total Feed Quantity Constraint:

   • $x + 0.5y \geq 5$

2. Digestible Protein Requirement:

   • $80x + 200y \geq 560$

3. Calcium Requirement:

   • $x + 8y \geq 20$

4. Non-Negativity Constraints:

   • $x \geq 0$
   • $y \geq 0$

Summary of the Model

The linear programming model is as follows:

$\text{Minimize } Z = 3x + 5y$

Subject to: $x + 0.5y \geq 5$ $80x + 200y \geq 560$ $x + 8y \geq 20$ $x \geq 0, \; y \geq 0$

This LP model can be solved using optimization methods to find the values of $x$ and $y$ that minimize the cost while satisfying all nutritional requirements.

Would you like further details on solving this model, or have any specific questions about the setup?

Further Questions to Consider:

1. How would the model change if the cost of feed increased?
2. What if the protein requirement increased by 10%?
3. How would you incorporate more types of feed into this model?
4. What impact would changes in calcium requirements have on the cost?
5. How can sensitivity analysis help in understanding the model better?

Tip:

Using a tool like Excel Solver, MATLAB, or Python's SciPy library can simplify solving this linear programming problem.