polynomial functions, yield, maximization

Question

suppose it is known that if 65 mango trees are planted in a certain lot, the average yield per tree will be 1500 mangoes per year. for each additional tree planted in the same lot, the annual yield per tree drops by 20 mangoes. how many mango trees should be planted in order to produce maximum crop of mangoes per year?

Answer 1

The yield per tree depends on the number of trees.    Let y(N) be the yield per tree if N trees are planted in the lot.

The formula for y(N)  can be seen to be  y(N) = 1500 - 20 (N-65).     The size of the crop,  S, is given by:

 

 S(N) = N y(N) = 2800 N  - 20 N² 

 

This is the objective function to be maximized.

 

  This can be viewed as a quadratic function (standard form:  f(x) = a x² + bx +c)  with a = -20, b = 2800 , c =0

 

 Since it is a downward opening quadratic, the maximum will be located at the vertex.  The formula for the location of the x coordinate of the vertex is   x_v =  -b/(2a).  

 

 This yields  N  =   - 2800 /(-40) =  70