Question

A farmer can buy four types of plant food. Each barrel of mix A contains 20 pounds of phosphoric acid, 50 pounds of nitrogen, and 20 pounds of potash, each barrel of mix B contains 20 pounds of phosphoric acid, 75 pounds of nitrogen, and 30 pounds of potash, each barrel of mix C contains 20 pounds of phosphoric acid, 25 pounds of nitrogen, and 30 pounds of potash, and each barrel of mix D contains 40 pounds of phosphoric acid, 25 pounds of nitrogen, and 50 pounds of potash. Soil tests indicate that a particular field needs 480 pounds of phosphoric acid, 550 pounds of nitrogen, and 640 pounds of potash. How many barrels of each type of food should the farmer mix together to supply the necessary nutrients for the field?

Let x₁ be the number of barrels of Mix A, x₂ be the number of barrels of Mix B, x₃ be the number of barrels of Mix C, and x₄ be the number of barrels of Mix D. Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.

Answer 1

To solve the fertilizer mix problem, set up a system of linear equations based on the pounds of phosphoric acid, nitrogen, and potash in each barrel and the soil's requirements. Solve the system to find the number of barrels needed for each mix type.

Step-by-step explanation:

The student is asking for help with a linear algebra problem involving mixing different types of NPK fertilizers to meet soil nutrient requirements. Each barrel of mix has different amounts of phosphoric acid, nitrogen, and potash. The goal is to determine the number of barrels of each mix (A, B, C, and D) needed to achieve the desired amounts of these components in the soil. This problem can be solved by setting up a system of linear equations based on the information provided.

The equations based on the given information will be:

• 20x₁ + 20x₂ + 20x₃ + 40x₄ = 480 (phosphoric acid requirement)
• 50x₁ + 75x₂ + 25x₃ + 25x₄ = 550 (nitrogen requirement)
• 20x₁ + 30x₂ + 30x₃ + 50x₄ = 640 (potash requirement)

By solving the system of equations, we can find the values of x₁, x₂, x₃, and x₄ that satisfy all three equations, which will give the number of barrels of each fertilizer mix required.