Two years ago your orange orchard contained 50 trees and the

Two years ago your orange orchard contained 50 trees and the yield per tree was 80 bags of oranges. Last year you removed 10 of the trees and noticed that the yield per tree increased to 85 bags. Assuming that the yield per tree depends linearly on the number of trees in the orchard, what should you do this year to maximize your total yield?

Solution

Let x be the number of trees and T be the total yield and P be the yield per tree.

T(50) = 80*50 = 4000

P(50) = 80

T(40) = 85*40 = 3400

P(40) = 85

Since \"yield per tree depends linearly on the number of trees,\" we know that
P(x+k) = P(x) + ck
P(50) = P(40) + 10c
80 = 85 + 10c
c = -.5
so P(x) = 80 - .5x

We must then maximize T(x) = x(80-.5x) = 80x - .5x^2
Set the derivative to 0:
T\'(x) = 80 - x = 0
x = 80

So if you currently have 80 trees, you already have the optimal number of trees. Keep your 80 trees.