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------------------------------------------------ Nutrition Ruff, Inc. makes dog food out of chicken and grain. Chicken has 10 grams of protein and 5 grams of fat per ounce, and grain has 2 grams of protein and 2 grams of fat per ounce. A bag of dog food must contain at least 206 grams of protein and at least 156 grams of fat. If chicken costs 9¢ per ounce and grain costs 1¢ per ounce, how many ounces of each should Ruff use in each bag of dog food to minimize cost?

The problem is an example of linear programming. It involves constructing linear inequalities representing the protein and fat requirements, graphing them together to find the feasible region, then identifying the combination of chicken and grain that minimizes cost while staying in this region.

Explanation:

This problem is an example of a linear programming problem where the goal is to minimize the cost while satisfying the nutritional requirements. Let's denote chicken as x and grain as y. We want to minimize the cost, which is 9x (cost of chicken) plus y (cost of grain), while maintaining nutritional balance. The nutritional requirement puts constraints on x and y:

For protein: 10x + 2y ≥ 206
For fat: 5x + 2y ≥ 156

To solve this, graph the inequalities, find the feasible region, and locate the vertices of the feasible region. Then apply the objective function (9x + y) to these vertices to find the minimum cost. The point with the lowest value from the function is the optimal solution, and will give the cost-minimizing combination of chicken and grain. This specific solution would depend on the shape of the feasible region which comes from the constraints.

This is a common problem in operations research, on providing the least cost possible while meeting specified constraints, such as in this content loaded Nutrition Ruff, Inc. scenario where dog food is made out of chicken and grain.

Learn more about Linear Programming here:

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