SOLUTION: An orchard contains 40 apple trees, each of which yields about 500 apples over the growing season. The owner plans to add more trees, but each new tree will reduce the average yiel

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Question 906174: An orchard contains 40 apple trees, each of which yields about 500 apples over the growing season. The owner plans to add more trees, but each new tree will reduce the average yield per tree by about 10 apples over the season. How many trees should be added to maximize the total yield of apples? What would be the maximum yield for the apple orchard for the season?
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let +x+ = number of apple trees added
Let +y+ = the yield for the orchard
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+y+=++%28+40+%2B+x+%29%2A%28+500+-+10x+%29+
+y+=+20000+%2B+500x+-+400x+-+10x%5E2+
+y+=+-10x%5E2+%2B+900x+%2B+20000+
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Find +x%5Bmax%5D+ and then +y%5Bmax%5D+
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+x%5Bmax%5D+=+-b%2F%282a%29+
+a+=+-10+
+b+=+900+
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+x%5Bmax%5D+=+-900%2F%282%2A%28-10%29%29+
+x%5Bmax%5D+=+45+
and
+y%5Bmax%5D+=+-10%2A45%5E2+%2B+900%2A45+%2B+20000+
+y%5Bmax%5D+=+-20250+%2B+40500+%2B+20000+
+y%5Bmax%5D+=+40250+
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45 more trees will maximize yield
40,250 will be the maximum yield
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Here's the plot:
+graph%28+500%2C+500%2C+-30%2C+150%2C+-4000%2C+45000%2C+-10x%5E2+%2B+900x+%2B+20000+%29+
( note that +x=0+ doesn't mean no apples- it means no ADDITIONAL apples )