Expert: David Hemmer - 4/17/2007

Question
hallo sir can you help me? im from malaysia, i got an exercise but it's too difficult to solve, so i need you favour to solve this for me, ill glade to receive any answer from you, the question is there are 500 apple trees in an orchard, each tree produces 8000 apples. for each additional tree planted in the orchard, the outpit per tree drops by 100 apples. how many trees should be added to the existing orchard in order to maximize the total output of trees? i need the step tin order to solve this problem sir.. thankyou

Answer
Let x be the number of trees. The total output per tree is

8000-(x-500)*100=8000-100x+50000=58000-100x

since this drops the output by 100 for each tree beyond 500.

Thus the total output for x trees is:

Output=x(58000-100x)=58000x-100x^2

To maximize this set the derivative equal to zero.

58000-200x=0

x=290.

This suggests trees should be removed, are you sure you gave me the right numbers? For instace if you add one tree then the output per tree drops to 7900 so you lose 50000 apples and only gain 7900.