Question 1180597: A farmer has available 300 hours of labour per week and 800 tons of fertilizer, and he has a maximum of 26 acres for strawberries and 37 acres for tomatoes. An acre of strawberries requires 10 hours of labour and 8 tons of fertilizer, whereas an acre of tomatoes requires 3 hours of labour and 20 tons of fertilizer. The profit from an acre of strawberries is $40,000 and the profit from an acre of tomatoes is $30,000. The farmer wants to know the number of acres of strawberries and tomatoes to plant to maximize profit.
a. Formulate a linear programming model for this problem. (7 marks)
b. Solve this model by using graphical analysis. (8 marks)
c. What would be the effect on the number of acres of strawberries and tomatoes to plant, and the maximum profit if the profit from an acre of strawberries was $50,000 instead of $30,000?
Answer by ankor@dixie-net.com(22740)   (Show Source):
You can put this solution on YOUR website! A farmer has available 300 hours of labour per week and 800 tons of fertilizer, and he has a maximum of 26 acres for strawberries and 37 acres for tomatoes.
An acre of strawberries requires 10 hours of labour and 8 tons of fertilizer, whereas an acre of tomatoes requires 3 hours of labour and 20 tons of fertilizer.
The profit from an acre of strawberries is $40,000 and the profit from an acre of tomatoes is $30,000. The farmer wants to know the number of acres of strawberries and tomatoes to plant to maximize profit.
:
a. Formulate a linear programming model for this problem. (7 marks)
Let x = no. of acres of strawberries
let y = no. of acres of tomatoes
:
The acre restraints
x =< 26
y =< 37
the labor equation, arrange both so we can graph it
10x + 3y = 300
3y = -10x + 300
y = (-10/3)x + 100
the amt of fertilizer equation
8x + 20y = 800
20y = -8x + 800
x = = -.4x + 40
:
b. Solve this model by using graphical analysis. (8 marks)
green is fertilizer equation, red is labor, and blue is the tomato constraint
A vertical line at x=26, should be there for the strawberry constraint, but unable to draw that
intersection occurs at x=21, y=30
21 acres of strawberries yields, 21*40000 = $840000
30 acres of tomatoes yields, 30 * 30000 = $900000
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The maximum profit with these two crops : $1740000
:
c. What would be the effect on the number of acres of strawberries and tomatoes to plant, and the maximum profit if the profit from an acre of strawberries was $50,000 instead of $30,000?---wasn't it $40000.
:
Using the graph, If you planted the max, 26 acres of strawberries,
you could only plant 13 acres of tomatoes:
26(50000) + 13(30000) = $169000 profit
however using the same acreage 21 and 30
21(50000) + 30(30000) = 1140000 would be max profit
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