Question

A fammer can buy two types of plant food, mix A and mx8. Each abic yard of mix A contains 20 pounds of phosphoric aod, 30 pounds of nitrogen, and 5 pounds of polash. Each cubic yard of mir 8 contins 10 pounds of phosphoric acid, 30 pounds of ntrogen, ond 10 pounds of polahh. The minimum monthy requirements ave 500 pounds of phesphoric acid, 990 pounds of ntrogen, and 210 . pounds of potash. If mixA conts 530 per cubic yard and mix B conts $40 per oubic yoed, how mary cubic yards of each mx should the femer biend to meet the minimum montly requirements at a minimum cont? What is this cost?

Answer 1

To meet the minimum monthly requirements at a minimum cost, the farmer should buy 9 cubic yards of mix A and 17.5 cubic yards of mix B. The total cost of the purchase would be $970.

Step-by-step explanation:

To determine the number of cubic yards of each mix the farmer should buy, we can set up a system of equations based on the minimum monthly requirements:

Let x = number of cubic yards of mix A to buy

Let y = number of cubic yards of mix B to buy

According to the requirements, the system of equations is:

20x + 10y = 500 (phosphoric acid)

30x + 30y = 990 (nitrogen)

5x + 10y = 210 (potash)

To solve this system, we can multiply the first equation by 2 and subtract the second equation from it to eliminate y. Then, we can multiply the first equation by 2 and subtract the third equation from it to eliminate x.

After solving, we find that x = 9 and y = 17.5. Therefore, the farmer should buy 9 cubic yards of mix A and 17.5 cubic yards of mix B to meet the minimum monthly requirements.

The total cost of the purchase can be calculated by multiplying the amount of each mix by its respective cost and adding them together:

Cost = (9 * $30) + (17.5 * $40) = $270 + $700 = $970. Therefore, the total cost is $970.