An animal food must provide 54 units of vitamins and 61 calories per serving. One gram of soybean meal provides 2.5 units of vitamins and 5 calories. One gram of meat byproducts provides 4.5 units of vitamins and 3 calories. One gram of grain provides 5 units of vitamins and 10 calories. A gram of soybean meal costs 7 cents, a gram of meat byproducts costs 9 cents, and a gram of grain costs 10 cents.

(a) What mixture of these three ingredients will provide the required vitamins and calories at minimum cost?
(b) What is the minimum cost?

(a) Set up the initial tableau for the dual maximum problem. Fill in the missing entries.

[tex]
\[
\begin{array}{rrrrrrr}
x_1 & x_2 & s_1 & s_2 & s_3 & z \\
2.5 & 5 & 1 & 0 & 0 & 0 & 7 \\
4.5 & 3 & 0 & 1 & 0 & 0 & 9 \\
5 & 10 & 0 & 0 & 1 & 0 & 10 \\
\hline
-54 & -61 & 0 & 0 & 0 & 1 & 0
\end{array}
\]
[/tex]

There is more than one optimal basic solution to this problem. The answer depends on whether the tie in the minimum ratio rule is broken by pivoting on the second row or third row of the dual.

- When pivoting on the second row of the dual, the mixture would contain [tex]\(\square\)[/tex] grams of soybean meal, [tex]\(\square\)[/tex] grams of meat byproducts, and [tex]\(\square\)[/tex] grams of grain.
- When pivoting on the third row of the dual, the mixture would contain [tex]\(\square\)[/tex] grams of soybean meal, [tex]\(\square\)[/tex] grams of meat byproducts, and [tex]\(\square\)[/tex] grams of grain.

(Round to the nearest tenth as needed.)

To solve the problem of finding the most cost-effective mixture of soybean meal, meat byproducts, and grain to meet the dietary requirements, we first analyze the given information:

### Requirements and Contributions:
1. Vitamin Requirement: 54 units
2. Calorie Requirement: 61 calories

Each ingredient provides:
- Soybean Meal: 2.5 units of vitamins and 5 calories per gram
- Meat Byproducts: 4.5 units of vitamins and 3 calories per gram
- Grain: 5 units of vitamins and 10 calories per gram

Costs per gram are:
- Soybean Meal: 7 units
- Meat Byproducts: 9 units
- Grain: 10 units

### (a) Mixture for Minimum Cost
To find the optimal mixture, we need to use the strategic information provided:

- Soybean Meal: 0 grams
- Meat Byproducts: 7.8 grams (rounded to the nearest tenth)
- Grain: 3.8 grams (rounded to the nearest tenth)

This mixture ensures that the vitamins and calorie requirements are met at the lowest possible cost while fulfilling the constraints provided by each ingredient's contribution to vitamins and calories.

### (b) Minimum Cost
The minimum cost of this optimal mixture is 108 units.

### Explanation:
The result suggests that using 7.8 grams of meat byproducts and 3.8 grams of grain satisfies both the vitamin and calorie requirements effectively. The calculated minimum cost reflects the efficiency of this particular combination in meeting the dietary needs at the lowest expense.

This step-by-step breakdown provides clarity in understanding how the optimal solution is achieved under given nutritional and cost constraints.