Orchard Tree Word Problem

Question

An orchard tree contains 200 apples, each of them yields about 500 apples over the growing season, the owner adds more trees, but the new tree will reduce the average yield per trees by about 9 apples over the season. How many trees should be added to maximize the total yields of apples? What would be the maximum yield for apple orchard for the season? 

 

 

How many trees should be added to maximize the total yields of apples

 

 

What would be the maximum yield for apple orchard for the season?

Answer 1

200 apples on a tree, yield 500 apples for the season

add another tree and each tree yields 9 less or 491 apples per tree for the season

Yield = 500 + (500-9)T where T=additional trees

Y = 500 +491T

1 more tree means Y=(500-9)x2

2 more trees means Y= (500-9(2))x3

3 more trees means Y= (500-3(9))x4

n more trees means Y= (500-n(9))x(n+1)

Y= (500-9n)(n+1)

=500 -9n+500n -9n^2

= -9n^2 +491n + 500

take the derivative and set = 0

-18n +491 = 0

18n=491

n= 491/18= 27.2727... = about 27 more trees to maximize apples

Y = (500-9n)(n+1) = 500-9(27))(27+1) = 257(28) = 7,196 apples= maximum output of apples with 27 more trees