An animal shelter mixes two brands of dog food.Brand X costs $29 per bag and contains two units of nutritional element A,two units of element B,and two units of element C.Brand Y costs $22 per bag and contains one unit of nutritional element A,nine units of element B,and three units of element C.The minimum required amounts of nutrients A,B,and C are 12 units,36 units,and 24 units,respectively.What is the optimal number of bags of each brand that should be mixed? What is the optimal cost?

Question

An animal shelter mixes two brands of dog food.Brand X costs $29 per bag and contains two units of nutritional element A,two units of element B,and two units of element C.Brand Y costs $22 per bag and contains one unit of nutritional element A,nine units of element B,and three units of element C.The minimum required amounts of nutrients A,B,and C are 12 units,36 units,and 24 units,respectively.What is the optimal number of bags of each brand that should be mixed? What is the optimal cost? ​

Quizplus · Accepted Answer

To meet the minimum required amounts of nutrients A (12 units), B (36 units), and C (24 units) at the lowest cost, we can set up equations based on the given information. Brand X provides 2 units of A, B, and C each, while Brand Y provides 1 unit of A, 9 units of B, and 3 units of C. By mixing 3 bags of Brand X and 6 bags of Brand Y, we get exactly 12 units of A (3*2 + 6*1), 36 units of B (3*2 + 6*9), and 24 units of C (3*2 + 6*3), meeting the minimum requirements exactly. The cost for this mix is 3*$29 (for Brand X) + 6*$22 (for Brand Y) = $87 + $132 = $219, which is the lowest cost option that meets all the nutrient requirements.

An Animal Shelter Mixes Two Brands of Dog Food