Question

A farmer can buy two types of plant​ food, mix a and mix

b. each cubic yard of mix a contains 20 pounds of phosphoric​ acid, 30 pounds of​ nitrogen, and 5 pounds of potash. each cubic yard of mix b contains 10 pounds of phosphoric​ acid, 30 pounds of​ nitrogen, and 10 pounds of potash. the minimum monthly requirements are 440 pounds of phosphoric​ acid, 990 pounds of​ nitrogen, and 200 pounds of potash. if mix a costs ​$20 per cubic yard and mix b costs ​$30 per cubic​ yard, how many cubic yards of each mix should the farmer blend to meet the minimum monthly requirements at a minimum​ cost? what is this​ cost?

Answer 1

To find the minimum cost of blending the plant food mixes, set up a system of equations to calculate the quantities of each mix needed. Solve the equations to find the values of x and y, then calculate the cost using the mix prices.

Step-by-step explanation:

To find the minimum cost of blending the plant food mixes, we need to set up a system of equations and solve for the quantities of each mix needed. Let's let x be the number of cubic yards of mix a and y be the number of cubic yards of mix b that the farmer needs.

The total pounds of phosphoric acid required from mix a and mix b should equal the minimum monthly requirement of 440 pounds. This gives us the equation: 20x + 10y = 440.

The total pounds of nitrogen required from mix a and mix b should equal the minimum monthly requirement of 990 pounds. This gives us the equation: 30x + 30y = 990.

The total pounds of potash required from mix a and mix b should equal the minimum monthly requirement of 200 pounds. This gives us the equation: 5x + 10y = 200.

Solving this system of equations will give us the values of x and y. Once we have these values, we can calculate the cost by multiplying the quantity of each mix by their respective prices and adding them together.