Question

A farmer plans to plant wheat and rye. However, he has only $1200 to spend. Each acre of wheat costs $200 to plant and each acre of rye costs $100 to plant. Moreover, the farmer has to get the planting done in 12 days before it's too late. It takes one day to plant an acre of wheat and 2 days to plant an acre of rye. If the profit is $500 per acre of wheat and $300 per acre of rye. How many acres of each should be planted to maximize profits

Answer 1

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Answer:

4 acres each of wheat and rye

Explanation:

Let x and y represent acres of wheat and rye, respectively. Then the constraints of the problem are ...

200x +100y ≤ 1200 . . . . . . cost constraint

x + 2y ≤ 12 . . . . . . . . . . . time constraint

The objective is to maximize profit (p):

p = 500x +300y

The constraints can be graphed (see attached). The profit function will be maximized at one of the vertices of the "feasible region," the portion of the graph that satisfies the constraints. The vertices are (x, y) = (0, 6), (4, 4), and (6, 0). The associated profit values are $1800, $3200, and $3000.

Maximum profit is obtained when 4 acres of wheat and 4 acres of rye are planted.