Posted by **Q** on Tuesday, March 25, 2014 at 8:09pm.

There are 50 apple trees in an orchard, and each tree produces an average of 200 apples each year. For each additional tree planted within the orchard, the average number of apples produced drops by 5. What is the optimal number of trees to plant in the orchard?

I mostly need help getting an equation and defining a variable

- Calculus -
**Steve**, Tuesday, March 25, 2014 at 8:16pm
the number of apples is yield/tree * # trees.

With x trees, yield per tree is 200 - 5(x-50) for x > 50

So, total crop is

c(x) = x(200-5(x-50)) = x(450-5x) = 450x - 5x^2

c'(x) = 450-10x

c' = 0 at x=45

So, the max yield is achieved with 45 trees

- Calculus -
**bobpursley**, Tuesday, March 25, 2014 at 8:28pm
number apples=average*number trees

let x be the nubmer of trees 50<x<inf

number apples=(200-(x-50)*5)(x) where x is the number of trees, 50<x<infinity

N=200x-5x^2 +250x

dN/dx=0=200-10x+250

10x=450

x=45

but x>50, so look at optimal

check x=50 N=50*200=10000 apples

check x=51 N=(51)(195)=9945

so indeed, the optimal is 50.

- Calculus -
**bobpursley**, Tuesday, March 25, 2014 at 8:29pm
now if you are allowed to remove trees, and for each tree removed, the average goes up by 5, then the optimal is to remove five trees.

- Calculus - good catch bob -
**Steve**, Tuesday, March 25, 2014 at 8:35pm
Dang - forgot the restriction on the domain.

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