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Question 593978: A small farm has 70 acres of space to plant. The farmer wants to grow corn and green beans. It costs $60 per acre to grow the corn and $30 per acre to grow the green beans. The farmer has $1800 to use for planting the vegetables. It takes the farmer 3 days to plant each acre of corn and 4 days to plant each acre of green beans. There are 120 days available to plant the vegetables. When the farmer sells the vegetables, he can expect to make a profit of $180 per acre for the corn and $100 per acre for the green beans. How many acres should the farmer allow for each vegetable in order to maximize profit?
What are the equations I should use to solve this problem?
Answer by ankor@dixie-net.com(22740)   (Show Source):
You can put this solution on YOUR website! A small farm has 70 acres of space to plant.
The farmer wants to grow corn and green beans.
It costs $60 per acre to grow the corn and $30 per acre to grow the green beans.
The farmer has $1800 to use for planting the vegetables.
It takes the farmer 3 days to plant each acre of corn and 4 days to plant each acre of green beans.
There are 120 days available to plant the vegetables.
When the farmer sells the vegetables, he can expect to make a profit of $180 per acre for the corn and $100 per acre for the green beans.
How many acres should the farmer allow for each vegetable in order to maximize profit?
:
Let x = no. of acres for corn
Let y = no. of acres for beans
:
Write an equation for each constraint, put each equation the slope intercept for for graphing
:
The Acerage equation
x + y =< 70
y = -x + 70, (Red)
:
Cost equation
60x + 30y =< 1800
30y = -60x + 1800
y = -2x + 60, Green
:
Time equation
3x + 4y =< 120
4y = -3x + 120
y = -.75x + 30 (blue)
:
Obviously, all values are positive
x => 0
y => 0
;
Graph all 3 equations on the same grid
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From the graph we can see an intersection at x = 24, y = 12
Check and you can see these values will give us a cost of $1800 and time of 120 days
We can ignore the acreage equation, only 36 acres can be planted
:
"When the farmer sells the vegetables, he can expect to make a profit of $180 per acre for the corn and $100 per acre for the green beans."
Profit:
24(180) + 12(100) =
4320 + 1200 = $5520 max profit
:
But another corner is x=30, y=0; 30 acres of corn and no beans
It also will cost $1800 and but only 90 days to plant
Profit: 30(180) = $5400, slightly less, but less time required
:
You can check the 3rd corner. x = 0; y = 30, 30 acres of beans and no corn
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