How many trees should be planted on an acre in order to get the highest yield?

Question

An apple farm yields an average of 30 bushels of apples per tree when 20 trees are planted on an acre of ground. Each time 1 more tree is planted per acre, the yield decreases by 1 bushel (bu) per tree as a result of crowding. How many trees should be planted on an acre in order to get the highest yield?

Answer 1

20 trees yields 30 bushels per tree

for each tree planted  the yield decreases by 1 bushel per tree

let x=# of trees planted to get highest yield

(20+x)(30-x)=f(x)

600+30x-20x-x²=f(x)

600+10x-x²=f(x)

(-b/2a)=(-10/-2)=5 trees, or taking the derivative...

-2x+10=0

-2x=-10

x=5 trees

you can also complete the square

x²-10x-600=0

x²-10x+25-600-25=0

(x-5)²-625=0

5 trees and maximum yield of 625 bushels

look at it like this also...

bushels          trees

30                  20

29                  21

28                  22

27                  23

26                  24

25                  25

24                  26

23                  27

you see that 25*25=625 and 26*24=624 

 

Answer 2

First you must come up with an equation of the yield

 

Yield = Bushels per tree * number of trees

 

Bushels per tree = 30 when the number of trees is 20 and decreases by 1 for each additional tree

 

x = number of trees

 

Bushels per tree = 30 when x = 20 so bushels per tree = (50 - x)    (50-20) = 30

 

Yield = (50 - x) * x

 

Yield = 50x - x²

 

take the derivative  and you get 50 - 2x

 

set this to zero to find the maximum  50 - 2x = 0    and 2x = 50 and x =25

 

Maximum yield will be given with 25 trees

 

you can test 24 and 26 to see that the yield is less

 

yield = (50-24)24 = 624

 

yield = (50-25)25 = 625

 

yield = (50-26)26 = 624

Answer 3

Yield = (Number of Trees per acre)*(Number of bushels per tree)

 

Yield = (22 + x)*(36 - x)

Yield = 22*36 + 36x - x²

 

To find the maximum yield, take the first derivative of Yield wrt x, set it to zero, and solve for x:

 

d(Yield)/dx = 36 - 2x

0 = 36 - 2x

2x = 36

x=18

 

Optimal trees per acre = 22 + 18 = 40