Question

An orange grower find that she gets an average yield of 40 bushels per tree when she plants 20 trees on an acre of ground. Each time she adds a tree to an acre, the yield per tree decreases by 1 bushel, due to congestion.

How many trees per acres should she plant for maximum yield?

Answer 1

30 trees/acre

Explanation:

Let n = the number of trees added to 1 acre

Let Y(n) = the yield in bushels/acre

Yield in bushels/acre = [bushels/tree] x [trees/acre]

Y(n) = (40-n)*(20+n)

= 800 - 20n + 40n - n^2

= n^2 + 20n + 800 ---------------------(1)

The n-value of the vertex ( which is a peak ) is given by the formula:

n(max) = -b/2a

Putting values from equation (1) gives us

n(max) = 10

The grower started with 20 tree/acre and adds 10 more for max yield, so she should plant: 30 trees/acre

and

Maximum yield is 900 bushels/acre.