Question

A farmer has 500 acres of arable land on which he wants to plant potatoes and corn. The farmer has $40,000 available for planting and $30,000 for fertilizer. Planting 1 acre of potatoes costs $90, and planting 1 acre of corn costs $50. Fertilizer costs $30 for 1 acre of potatoes and $80 for 1 acre of corn. (a) Find a system of inequalities that describes the number of acres of each crop that the farmer can plant with the available resources. (Let x and y be the numbers of acres of potatoes and corn, respectively.)

Answer 1

The system of inequalities is: $90x + $50y ≤ $40,000 and $30x + $80y ≤ $30,000.

Step-by-step explanation:

To solve the problem faced by the farmer, we need to develop a system of inequalities that encapsulates the constraints on planting and fertilizing. Let x represent the acres of potatoes and y represent the acres of corn to be planted. We have four constraints based on the given information:

1. The total acreage cannot exceed 500, so x + y ≤ 500.
2. For planting costs: planting potatoes costs $90 per acre and corn $50 per acre, and the farmer has $40,000 in total. Therefore, the inequality for planting costs is 90x + 50y ≤ 40,000.
3. For fertilizer costs: potatoes require $30 per acre and corn requires $80 per acre, with $30,000 available. Thus, the inequality for fertilizer is 30x + 80y ≤ 30,000.
4. Since the farmer cannot plant a negative area of land, we also have x ≥ 0 and y ≥ 0.

Combining these constraints forms the system of inequalities that describes the situation:

• x + y ≤ 500
• 90x + 50y ≤ 40,000
• 30x + 80y ≤ 30,000
• x ≥ 0
• y ≥ 0

Answer 2

90x+ 50 y ≤ $40,000------ A

30x+ 80y ≤ $ 30,000------- B

x + y ≤ 500 --- C

Step-by-step explanation:

Let x and y be the numbers of acres of potatoes and corn, respectively.

Then according to the given conditions

90x+ 50 y ≤ $40,000------ A

30x+ 80y ≤ $ 30,000------- B

x + y ≤ 500 --- C

where both x and y equal to or greater than zero and cannot take a value less than zero.

x + y ≤ 500 --- C

gives us an idea that the total arable land is less or equal to 500 acres.

90x+ 50 y ≤ $40,000------ A

tells us that the total cost of planting potatoes and corn is less or equal to $ 40,000

30x+ 80y ≤ $ 30,000------- B

tells us that the total cost of fertilizers for potatoes and corn is less or equal to $ 30,000

In each of these x and y cannot have negative values or less than zero.