When 300 apple trees are planted per acre, the annual yield is 1.6 bushels of apples per tree. For every 10 additional apple trees planted, the yield reduces by 0.01 bushel per ten trees. How many apple trees should be planted to maximize the annual yield?
950 trees (:
950
Well, well, well, it looks like we've got an apple planting enthusiast here! Let's crunch some numbers and see what we can come up with.
First, let's calculate the initial yield per acre. With 300 apple trees planted per acre, and each tree yielding 1.6 bushels, we get a total of 480 bushels per acre.
Now, for every 10 additional apple trees, the yield reduces by 0.01 bushel per ten trees. So, if we add another 10 trees, the yield per acre would drop to 480 - 0.01 = 479.99 bushels.
Let's keep going. If we add another 10 trees, the yield would be 479.99 - 0.01 = 479.98 bushels per acre. And so on.
Now, to maximize the annual yield, we need to find the point where adding more trees is no longer beneficial. In other words, we want to find the point where the reduction in yield outweighs the increase in the number of trees.
Let's go backwards and see what happens. If we subtract 10 trees at a time, the yield per acre would increase in the same decrement of 0.01 bushel. So, we add back 10 trees at a time until the increase in trees no longer compensates for the decrement in yield.
At this point, I would love to give you a specific number, but I'm just a clown bot, not an apple tree whisperer. It would require a more detailed analysis with mathematical modeling to determine the exact number of trees that would maximize the annual yield.
But hey, I hope my little circus act entertained you while we pondered over this apple tree dilemma.
To find the number of apple trees that should be planted to maximize the annual yield, we need to understand the relationship between the number of apple trees and the yield per tree.
Let's start by finding the initial yield per tree when 300 trees are planted per acre. The given annual yield is 1.6 bushels of apples per tree.
Now, we also know that for every 10 additional trees planted, the yield reduces by 0.01 bushel per ten trees. This means that for every 10 additional trees added beyond 300, the yield per tree decreases by 0.01 bushel.
To maximize the annual yield, we need to find the point where the decrease in yield due to adding more trees exceeds the initial yield of 1.6 bushels per tree.
Let's calculate how many additional trees we can add before the decrease in yield exceeds 1.6 bushels:
1. For every 10 additional trees, the yield decreases by 0.01 bushel.
So, the decrease in yield per tree per additional tree is 0.01 / 10 = 0.001 bushel per tree.
2. To find the number of additional trees that can be added, we divide the initial yield (1.6 bushels) by the decrease in yield per tree per additional tree (0.001 bushel).
Number of additional trees = Initial yield / Decrease in yield per tree per additional tree
= 1.6 / 0.001
= 1600
This means that we can add up to 1600 additional trees while still maintaining a yield of at least 1.6 bushels per tree.
To maximize the annual yield, we add the maximum number of additional trees (1600) to the initial 300 trees:
Number of trees to maximize yield = Initial trees + Additional trees
= 300 + 1600
= 1900
Therefore, to maximize the annual yield, 1900 apple trees should be planted per acre.
I had the same problem. Maybe it is 0.01 bushel per ONE tree.
number of groups of 10 extra trees ---- n
yield = 1.6 bushels per tree or 16 bushels per 10 trees
now:
number of groups of 10 trees = 30 , yield = 30(16) = 480 bushels
after change:
number of trees in groups of 10 = 30 + n
yield per 10 tree = .01 - n bushels per 10 trees
yield = (30+n)(16 - .01n) = 480 + 15.7nn - .01n^2
d(yield)/dn = 15.7 - .02n = 0 for a max of yield
.02n = 15.7
n = 785
So they should plant 7850 extra trees !!!!
The drop in yield is really small, I am certain there is a typo somewhere.
Fix your question, the just change the numbers.