An apple orchard has an average yield of 36 bushels of apples per tree if tree density is 26 trees per acre. For each unit increase in tree density, the yield decreases by 2 bushels per tree. How many trees per acre should be planted to maximize the yield?
_______tree/acre
To maximize the yield of the apple orchard, we need to find the tree density that results in the highest average yield per tree.
Let's start by defining the variables:
- T = Number of trees per acre
- Y = Average yield per tree (in bushels)
We know that the yield decreases by 2 bushels per tree for each unit increase in tree density. Therefore, the average yield per tree (Y) can be calculated as follows:
Y = 36 - 2(T - 26)
We want to find the tree density that maximizes the yield, so we need to find the maximum value of Y. To do this, we can take the derivative of Y with respect to T and set it equal to zero.
dY/dT = -2
Setting the derivative equal to zero, we have:
-2 = 0
Since the derivative is a constant and not dependent on T, there is no maximum or minimum point for the yield with respect to tree density.
However, we know that the highest average yield per tree is 36 bushels, which occurs when the tree density is 26 trees per acre.
Therefore, to maximize the yield in the orchard, the number of trees per acre should be 26 trees/acre.