Answer:
48
Explanation:
You want to know the number of trees per acre that should be planted to maximize yield if 27 per acre yields 70 pounds of cherries per tree, and the yield goes down by 1 pound per tree for each additional tree planted.
Yield
The total yield per acre is the number of trees multiplied by the yield per tree. If we let x represent the number of trees per acre, then the yield per tree is ...
(70) -(x -27) . . . . . pounds per tree
and the total yield per acre is ...
y = x(97 -x)
We recognize this equation as that of a parabola with zeros at x=0 and x-97. The vertex is on the line of symmetry halfway between these zeros, so is ...
ymax = (0 +97)/2 = 48.5
The yield per acre will be maximized by planting 48 trees per acre.
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Additional comment
The yield function is symmetrical about 48.5 trees, so 48 trees will give the same yield as 49 trees. There's no benefit to planting that 49th tree.
In fact, the total yield goes up by only 2 pounds for planting the 48th tree, so one wonders how that relates to the cost of planting and caring for that additional tree.
In Oregon, a "high density" sweet cherry orchard may have around 340 trees per acre. At that density, yield is about 41 pounds per tree. Due to costs of establishing and maintaining such an orchard, it does not begin to have a positive return on investment for about 16 years into the 25-year life of the trees.
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