Oz makes lion food out of giraffe and gazelle meat. Giraffe meat has 18 grams of protein and 36 grams of fat per pound, while gazelle meat has 36 grams of protein and 18 grams of fat per pound. A batch of lion food must contain at least 36,000 grams of protein and 54,000 grams of fat. Giraffe meat costs $4/pound and gazelle meat costs $8/pound. How many pounds of each should go into each batch of lion food in order to minimize costs What are the shadow costs of protein and fat Use the rounded numbers of pounds in your calculations. ?

Question

Oz makes lion food out of giraffe and gazelle meat. Giraffe meat has 18 grams of protein and 36 grams of fat per pound, while gazelle meat has 36 grams of protein and 18 grams of fat per pound. A batch of lion food must contain at least 36,000 grams of protein and 54,000 grams of fat. Giraffe meat costs $4/pound and gazelle meat costs $8/pound. How many pounds of each should go into each batch of lion food in order to minimize costs  What are the shadow costs of protein and fat  Use the rounded numbers of pounds in your calculations. ?

Quizplus · Accepted Answer

To minimize costs while meeting the protein and fat requirements, we can set up the following system of inequalities based on the given information:Let $x$ be the pounds of giraffe meat and $y$ be the pounds of gazelle meat.From the protein requirement:$$18x + 36y \geq 36,000$$From the fat requirement:$$36x + 18y \geq 54,000$$Minimizing cost:$$Cost = 4x + 8y$$Solving this system to meet the requirements at the lowest cost, we find that 1,333 pounds of giraffe meat and 333 pounds of gazelle meat satisfy the constraints with the minimum cost. The shadow costs, which represent the change in the objective function (total cost) per unit increase in the constraint (protein and fat), are calculated based on the constraints' binding values and the cost function. The correct shadow costs are 22 cents per gram of protein and 15 cents per gram of fat, indicating the additional cost per gram for each nutrient if the requirement is increased by one gram.

Oz Makes Lion Food Out of Giraffe and Gazelle Meat