# Calculus

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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 -

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 -

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 -

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 -

Dang - forgot the restriction on the domain.

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