Question 181787: Hi tutors!
I'm really stuck on this!!
an orchandist has 10 hectares of land available for growing Fuji apples and Nashi pears. She estimates that each hectare of apples will cost $5000 to establish, and each heactare of pears will cost $20 000 to establish. She is not able to spend more than $100 000 on establishing the orchard.
It is also estimated that the profit per hectare from apples will be $4000 and the profit per hectare from pears will be $8000.
What is the maximum profit the orchard can expect?
How many hectares of Fuji apples and Nashi pears should she plant to achieve this? Show your working clearly. You should explain the reason behind the answer you give.
This is a huge problem. Sorry about this tutors!! But would appreciate if you could solve it for me!!
Thanks!!
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! an orchandist has 10 hectares of land available for growing Fuji apples and Nashi pears.
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Land Inequality: A + P <= 10
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She estimates that each hectare of apples will cost $5000 to establish, and each heactare of pears will cost $20 000 to establish. She is not able to spend more than $100 000 on establishing the orchard.
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Cost: P <= 100,000/20,000 = 5
Cost: A <= 100,000/5000 = 20
Cost Inequality: 5000A + 20000P <= 100,000
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It is also estimated that the profit per hectare from apples will be $4000 and the profit per hectare from pears will be $8000.
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Object Function:
Profit = 4000A + 8000P
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What is the maximum profit the orchard can expect?
How many hectares of Fuji apples and Nashi pears should she plant to achieve this? Show your working clearly. You should explain the reason behind the answer you give.
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Simplify the Inequalities:
0<= P <=5
0<= A <=20
Land: A <= -P + 10
Cost: A <= -40P + 20
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Graphing: Let A be the vertical axis and P be the horizontal axis.
Draw a horizontal line at P=5
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Shade the area in the 1st quadrant that satisfies all of the inequalities.
Find the vertices of that shaded area relating A and P.
(0,5),(5,5),(5,0),(0,0)
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Test each of these in the Object Function:
Profit = 4000A + 8000P
(0,5): P = 0 + 40,000 = $40,000
(5,5): P = 20,000 + 40,000 = $120,000
(5,0): P = 5*4000+0 = $20,000
(0,0): P = 0 + 0 = $0
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Conclusion: Maximum profit comes when 5 hectares of apples are planted and
5 hectares of pears are planted.
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Cheers,
Stan H.
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