Problem 5 - Orange Grove Problem

Question

A farmer in Florida has an orange grove. In his grove there are 60 trees. Each tree ordinarily produces 500 oranges. The farmer wants to raise his overall orange production and knows that because of lost space and sunlight, every additional tree that he plants will cause a reduction of 5 oranges from each tree.

1. What is the maximum number of oranges that he will be able to produce in his grove?
2. How many trees will he need to reach this maximum? 

Show your work, using any strategy to derive your answer and communicate your reasoning.

Extension: Devise a scenario where the starting number of trees, the starting yield (number of oranges produced) for that number of trees, and the reduction per tree is different.

Answer 1

O= (60+t)(500-5t)= 3000+500t-300t- 5t^2

0ranges=3,000+200t-5t^2 where t=number of additional trees

he should add 20 trees for total 80 trees to get 5,000 Oranges =max possible

for 50 original trees and 400 oranges per tree originally, and 4 oranges less per extra tree:

O=(50+t)(400-4t)= 2000+200t-4t^2

then add 25 trees to get 75(300) = 22,500 Oranges = max

more generally

O=(T+t)(N-nt) where T=original trees, t=extra trees,N=original number of oranges per original Trees, n=reduction of oranges per tree

O(t)= TN+(N-n)t - nt^2

O'(t) = N-n -2nt = 0

t=(N-n)/2n = N/2n -1/2 = optimal number of extra trees

max Oranges = O(N/2n-1/2) = TN+(N-n)(N/2n-1/2) -n(N/2n-1/2)^2