Question

Farmer planting corn and soybeans: A farmer has a 320 acre farm on which she plants two crops: corn and soybeans. For each acre of corn planted, her expenses are $50 and for each acre of soybeans planted, her expenses are $100. Each acre of corn requires 100 bushels of storage and yields a profit of $60; each acre of soybeans requires 40 bushels of storage and yields a profit of $90. If the total amount of storage space available is 19,200 bushels and the farmer has only $20,000 on hand, how many acres of each crop should she plant in order to maximize her profit? What will her profit be if she follows this strategy?

Answer 1

Answer:

Let x represents acres of corn to be planted.

Let y represents acres of soybeans to be planted.

A farmer has a 320 acre farm on which she plants two crops. We get;

$x+y\leq 320$

Other equations as per scenario becomes:

$x\geq 0$

$y\geq 0$

$50x+100y\leq 20000$

$100x+40y\leq 19200$

We have to find the profit. So let the profit be p. We can give this by:

$p(x,y)=60x+90y$

When we graph these functions, we get the following.

(0,200) putting these in $p(x,y)=60x+90y$ we get profit as $1800

(140,130) , we get $20,100

(192,0) we get $11520

In the graph, we have points (240,180) but this is not possible as $x+y\leq 320$ so, we have (140,130) to get the maximum profit.