Word Problem Involving Maximum Yield

Question

When an apple orchard owner plants 65 trees on an acre of ground, he gets an average yield of 1500 apples per tree per year. For each additional tree planted per acre, the annual yield per tree drops by 20 apples. Determine the maximum yield.

Answer 1

 

 

y= (1500 - 20(t-65))t

 

t-65 is the amount of trees added

t is the amount of trees at any point

1500 minus 20(t-65) is how much the yield goes down for every tree added

 

this total times t, the amount of trees, will give us the new yield

 

To get the maximum, we set the derivative of the above function equal to 0.

 

 

y = 1500t - 20t² + 1300t

 

y = -20t² + 2800t

 

y^'= -40t + 2800 --> the derivative

 

0 = -40t + 2800 --> set derivative equal to 0 and solve for t

 

40t = 2800

t = 70

This is the amount of trees that will maximize the yield.

 

We then plug this into the original equation to get the actual maximum yield

 

y max = 98000 = maximum yield