An animal feed to be mixed from soybean meal and oats must contain at least 120 mathrm{lb} of protein, 24 mathrm{lb} of fat, and 10 mathrm{lb} of mineral ash. Each sack of soybeans costs 15 and contains 50 mathrm{lb} of protein, 8 mathrm{lb} of fat, and 5 mathrm{lb} of mineral ash. Each sack of oats costs 5 and contains 15 mathrm{lb} of protein, 5 mathrm{lb} of fat, and 1 mathrm{lb} of mineral ash. How many sacks of each should be used to satisfy the minimum requirements at minimum cost?

Question

An animal feed to be mixed from soybean meal and oats must contain at least 120 mathrm{lb} of protein, 24 mathrm{lb} of fat, and 10 mathrm{lb} of mineral ash. Each sack of soybeans costs  15 and contains 50 mathrm{lb} of protein, 8 mathrm{lb} of fat, and 5 mathrm{lb} of mineral ash. Each sack of oats costs  5 and contains 15 mathrm{lb} of protein, 5 mathrm{lb} of fat, and 1 mathrm{lb} of mineral ash. How many sacks of each should be used to satisfy the minimum requirements at minimum cost?

Accepted Answer

Step:1 ：Define the problem as a linear programming problem. The objective is to minimize the cost of the animal feed, which is the sum of the cost of the soybeans and the cost of the oats. The constraints are that the feed must contain at least 120 lb of protein, 24 lb of fat, and 10 lb of mineral ash.Step:2 ：Set up the coefficients of the objective function. The cost of a sack of soybeans is 15 and the cost of a sack of oats is 5, so the coefficients are ([15, 5]).Step:3 ：Set up the coefficients of the inequality constraints. Each sack of soybeans contains 50 lb of protein, 8 lb of fat, and 5 lb of mineral ash. Each sack of oats contains 15 lb of protein, 5 lb of fat, and 1 lb of mineral ash. So the coefficients are ([[-50, -15], [-8, -5], [-5, -1]]).Step:4 ：Set up the right-hand side of the inequality constraints. The feed must contain at least 120 lb of protein, 24 lb of fat, and 10 lb of mineral ash. So the right-hand side is ([-120, -24, -10]).Step:5 ：Set up the bounds for the variables. The number of sacks of soybeans and oats cannot be negative, so the bounds are ((0, None)) for both variables.Step:6 ：Solve the linear programming problem using the scipy.optimize.linprog function. The solution is approximately 1.846 sacks of soybeans and 1.846 sacks of oats.Step:7 ：Calculate the minimum cost. The cost of a sack of soybeans is 15 and the cost of a sack of oats is 5. So the minimum cost is approximately 36.92.Step:8 ：Final Answer: The minimum cost is approximately 36.92, achieved by purchasing approximately 1.846 sacks of soybeans and 1.846 sacks of oats. So, the final answer is (boxed{1.846 text{ sacks of soybeans}, 1.846 text{ sacks of oats}, 36.92}).

Answer

Step: 1 ：Define the problem as a linear programming problem. The objective is to minimize the cost of the animal feed, which is the sum of the cost of the soybeans and the cost of the oats. The constraints are that the feed must contain at least 120 lb of protein, 24 lb of fat, and 10 lb of mineral ash.Step: 2 ：Set up the coefficients of the objective function. The cost of a sack of soybeans is 15 and the cost of a sack of oats is 5, so the coefficients are ([15, 5]).Step: 3 ：Set up the coefficients of the inequality constraints. Each sack of soybeans contains 50 lb of protein, 8 lb of fat, and 5 lb of mineral ash. Each sack of oats contains 15 lb of protein, 5 lb of fat, and 1 lb of mineral ash. So the coefficients are ([[-50, -15], [-8, -5], [-5, -1]]).Step: 4 ：Set up the right-hand side of the inequality constraints. The feed must contain at least 120 lb of protein, 24 lb of fat, and 10 lb of mineral ash. So the right-hand side is ([-120, -24, -10]).Step: 5 ：Set up the bounds for the variables. The number of sacks of soybeans and oats cannot be negative, so the bounds are ((0, None)) for both variables.Step: 6 ：Solve the linear programming problem using the scipy.optimize.linprog function. The solution is approximately 1.846 sacks of soybeans and 1.846 sacks of oats.Step: 7 ：Calculate the minimum cost. The cost of a sack of soybeans is 15 and the cost of a sack of oats is 5. So the minimum cost is approximately 36.92.Step: 8 ：Final Answer: The minimum cost is approximately 36.92, achieved by purchasing approximately 1.846 sacks of soybeans and 1.846 sacks of oats. So, the final answer is (boxed{1.846 text{ sacks of soybeans}, 1.846 text{ sacks of oats}, 36.92}).

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