Wednesday, September 14, 2011

How to solve this story problem?

This is an optimization problem:



The book states that a florida citrus grower estimates that if 60 orange trees are planted, the average yield per tree will be 400 oranges. The average yield will decrease by 4 oranges per tree for each additional tree planted on the same acreage. How many trees should the grower plant to maximize the total yield?



Could you please show me how to solve this problem/set it up? Thanks!How to solve this story problem?
Okay, if you have 60 trees with 400 oranges each, you have 24,000 oranges. Now if you plant another tree, you lose 4 oranges a tree, so you have the new 396 oranges, but you've lost 240 oranges from the rest of the trees. You end up with a total of 24,156.



Now, if you plant another tree, you end up with an extra 392 oranges on the new tree, and you lose 244 from the other 61 trees, giving you a total of 24,304.



Notice how the number of oranges you get are decreasing, and the number of oranges you're losing is increasing?



A third tree gives you 388, loses you 248

A fourth tree gives you 384, loses you 252

A fifth tree gives you 380, loses you 256

A six tree gives you 376, loses you 260



The difference is coming together by eight each time, so with a difference of 116, it would be another 14 trees before you reach the maximum.



Now, 20 new trees would be 80 less oranges per tree, so you have 25,600 oranges and 80 trees.How to solve this story problem?
yield y(x) = (60+x)(400-4x) where x is the number of trees planted

y'(x) = -4(60+x) + (400-4x) = -240-4x+400-4x = 160-8x which is zero when x = 20 trees additionalHow to solve this story problem?
Yield per tree = y = 400 - 4(n-60) = 640 - 4n



Total Yield = Yield per tree * number of trees = y*n = Y = 640n - 4n虏

dY/dn = 0 = 640 - 8n =%26gt; n = 80



The grower should plant 80 trees on the acreage to maximize yield.

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