Maximum yield

Question

An apple orchard has an average yield of 32 bushels of apples per tree if tree density is 22 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?

_____trees/acre

Answer 1

Let x = number of additional trees to be planted per acre

 

Y(x) = yield (in bushels per acre) if x additional trees are planted

 

       = (number of trees)(yield per tree)

 

       = (22+x)(32-2x)    Note that x lies in the interval [0,16]

 

       = -2x² - 12x + 704

 

       = -4x - 12 = 0 when x = -3

 

So, there are no critical points in the interval [0,16].

 

Therefore, the maximum occurs at an endpoint.

 

Y(0) = 704 bushels

 

Y(16) = 0 bushels

 

Maximum yield when x = 0.  That is, to maximize yield, don't plant any more trees per acre.