Afarmer has a field of 70 acres in which he plants potatoes and corn. the seed for potatoes costs $20/acre, the seed for corn costs $60/acre, and the farmer has set aside $3000 to spend on seed. the profit per acre of potatoes is $150 and the profit for corn is $50 an acre. how many acres of each should the farmer plant?

The farmer should plant 30 acres of potatoes and 40 acres of corn.

In this question we must construct the following system of three equations:

(1)

(2)

, (3)

Where:

If we know that , , , , then we have the following system of equations:

(1)

(2b)

(3b)

By (1) in (2b):

(2c)

By (1) in (3b):

(3c)

And we eliminate to find a relationship between profits and seeding costs:

Linear functions do not have maxima and minima, then we must assume an arbitrary value for to obtain : ()

By (3c):

By (1):

The farmer should plant 30 acres of potatoes and 40 acres of corn.

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