Question

there are 50 apple trees in an orchard. each tree produces 800 apples. for each additional tree planted in the orchard, the output per tree drops by 10 apples. how many trees should be added to the existing orchard in order to maximize the total output of trees?

Answer 1

Let n = number of additional trees planted

Let t = total number of trees ............ t = 50 + n
Let a = number of apples per tree ........... a = 800 - 10n

Let p = total orchard production

Total production = number of trees * number of apples per tree
p(n) = (50 + n) (800 - 10n)
p(n) = -10n^2 + 300n + 40,000

p'(n) = -20n + 300
p''(n) = -20 < 0 for all n

p'(n) = 0
20n = 300
n = 15

p'(15) = 0 and p''(15) < 0
Therefore production is maximized when n = 15
p(15) = (50 + 15) (800 - 10*15) = 65 * 650 = 42,250

To maximize output of apples, 15 trees additional trees should be planted.
Total output of apples will then be 42,250.

http://answers.yahoo.com/question/index?qid=20101102203324AAY85wZ