An apple orchard now has 60 trees planted per hectare and the average yield is 500 apples per tree.

Question

An apple orchard now has 60 trees planted per hectare and the average yield is 500 apples per tree. For each additional tree planted per hectare, the average yield per tree is reduced by approximately five apples. How many trees per hectare will give the largest crop of apples?

Answer 1

The crop of the orchard is calculated by multiplying the number of trees per hectare by the average yield per tree:

1. If the number of additional trees per hectare is x, then the number of trees per hectare is 60 + x.
2. If the average yield per tree decreases by 5 for each additional tree, the average yield per tree is 500 - 5x.
3. The crop is therefore given by the function, f(x) = (60 + x)(500 - 5x) = 30,000 + 200x - 5x²

This is a concave down quadratic function with one stationary point, the maximum.

You can differentiate the function, put the derivative equal to zero, and solve for x. This will give you the x-value of the maximum, or the number of additional trees that give the largest crop.

f'(x) = 200 - 10x

200 - 10x = 0

x = 20 additional trees

Therefore, 60 + 20 = 80 trees per hectare will give you the largest crop.

Answer 2

Y(t) = (60 + t)(500 - 5t)

Convert to vertext form to determine maximum.