Question

An apple orchard contains 50 trees per hectare. The avergae yield per tree is 600 apples. If the trees are spaced more closely, when being planted, the yield per tree drops by 10 apples for each extra tree. Find the number of trees that should be planted per hectare in order to maximize the total yield.

Answer 1

55 trees per hectare.

Explanation:

An apple orchard contains 50 trees per hector. The average yield per tree is 600 apples.

If the trees are spaced more closely, when being planted, the yield per tree drops by 10 apples for each extra tree.

Let x extra tree is planted and then the average yield per tree reduces by 10x.

Therefore, yield as a function of x can be written as

Y(x) = (50 + x)(600 - 10x) = 30000 + 100x - 10x²

Therefore, condition for maximum yield is
`(dY(x))/(dx) = 0`

So, 100 - 20x = 0

⇒ x = 5

So, when the number of trees that should be planted per hectare is (50 + 5) = 55, then only the yield will be maximum. (Answer)