Question

The number of apples produced by each tree in an apple orchard depends on how densely the trees are planted. If n trees are planted on an acre of land, then each tree produces 900 - 9n apples. So the number of apples produced per acre is A(n) = n(900 - 9n) How many trees should be planted per acre to obtain the maximum yield of apples?

Answer 1

50 trees should be planted to obtain the maximum yield

Explanation:

In order to calculate the maximum yield of apples per acre, where

A(n) = n* (900 - 9 n)

A should be derived respect to n . The maximum value stands where the derivative equals 0. Thus,

A(n) = n* (900 - 9 n) = 900*n - 9* n²

therefore

dA/dn = 900 - 2*9*n = 0 → n= 900/(2*9) = 50 trees

the maximum yield of apples A will be

A(50) = 50* (900 - 9*50) = 22500 apples/acre when n=50 trees

Note

- Since the second derivative d²A/(dn)² = -18 < 0 we verify that is a maximum value

- n goes between 0 and 100 , with maximum at 50