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.

How to determine profits in a farm

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

Land occupied

(1)

Net profit

(2)

Seeding costs

, (3)

Where:

- Land occupied by potatoes, in acres. - Land occupied by corn, in acres. - Unit seeding costs for potatoes, in $ per acre. - Unit seeding cost for corn, in $ per acre. - Unit profit for potatoes, in $ per acre. - Unit profit for corn, in $ per acre. - Net profit, in $ - Seeding costs, in $

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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If you reduce, 2/10 = 1/5. So, it is LESS than 2/5